Exceptional polynomials

This page gives examples of exceptional polynomials according to the definition in my paper Distance to the discriminant. They are all homogeneous polynomials in three variables and therefore their zero locus are algebraic curves in the projective plane. These polynomials maximize the distance to the discriminant for the Bombieri norm, among polynomials of the same norm. As a consequence (see the paper), they can all be written as sums of powers of linear forms in the directions which correspond to critical points of the polynomial on the unit sphere with the minimum absolute critical value.

This page presents two kinds of polynomials: exact ones and numerical approximations. The former are normalized to be at distance one to the discriminant, while the latter are normalized to have Bombieri norm 1. In both cases, we give two expressions:

The original form $P$ arising from the construction of the curve.
The writing $P_{1}$ as a sum of power of linear forms.
For exact polynomials we have $P = P_{1}$ . For numerical polynomials, $\frac{∥ P - P_{1} ∥}{∥ P ∥}$ gives us an interesting error which tells us if the polynomial is optimized enough. For instance, if it is less than $\frac{dist (P, Δ)}{∥ P ∥}$ , we know that both polynomials have zero locus which are ambient-isotopic. When this is not the case, the optimization process to maximize the distance to the discriminant is not finished yet and the polynomial is marked with a 🔴 sign. We stopped optimisation when we got the best polynomial with coefficients represented as double (64 bits IEEE 754 representation) (marked as 🟢). When it is not the case, we will continue the optimisation (marked as 🟡). For the writing as a sum of power of linear forms, we give enough precision for the distance between the two polynomials to be accurate.

We provide images of the zero locus curves that can be zoomed and dragged. The curve is projected onto a disc shown in light grey using the stereographic projection $(x, y, z) \mapsto (\frac{x}{1 + z}, \frac{y}{1 + z})$ of the hemisphere $z > 0$ . We also highlight with a cross the critical points with the least critical absolute value on the unit sphere. These are the directions of the linear forms in the expression as a sum of linear forms to the power d.

Table of curves (see the lexicon below for explanation of the remarks)

Degree	Topology	Remark	Nb. of forms	$\frac{dist (P, Δ)}{∥ P ∥}$	$\frac{∥ P - P_{1} ∥}{∥ P ∥}$	Good

Lexicon

M curve: Algebraic curve with the maximum number of connected components for its degree: $M = \frac{(d - 1) (d - 2)}{2} + 1$ .
M-k curve: Algebraic curve with $M - k$ connected components ( $k$ below the maximum).
EM-k curve: Extremal M-k curve: no extra oval can be added without changing the relative position of the existing ovals or increasing the degree.
LM-k curve: Extremal M-k curve: no extra oval can be added locally without changing the relative position of the existing ovals. Locally means that the transformation is a deformation in the space of polynomials with the same degree using only one singular point which is not an isolated point of the curve.
Harnacks's M-curve: Refers to the 1876 construction of Harnack [Harnack 1876] reaching the maximum for any degree.
Hilberts's M-curve: Refers to the 1891 construction of Hilbert [Hilbert 1891] reaching the maximum with nests containing many ovals. The construction also gives the two EM-2 sextic curves presented on this page.
Gudkov's M-curve: Refers to the 1960s construction of Gudkov [Gudkov 1974] for the last missing sextic M-curve.

Bibliography

A. Harnack
“Über die Vieltheiligkeit der ebenen algebraischen Curven”
Mathematische Annalen, 10 (1876), 189–199.
D. Hilbert
“Ueber die reellen Züge algebraischer Curven”
Mathematische Annalen, 38 (1891), 115–138.
D. A. Gudkov
“The topology of real projective algebraic varieties”
Russian Mathematical Surveys, 29 (1974), 3–79. (Earlier version exists in russian)

Degree 2 to 5

Degree 2, ⟨1⟩ (M curve) 🟢

\begin{array}{rcl} P & = & x^{2} + y^{2} - z^{2} \\ = & x^{2} + y^{2} - z^{2} \\ ‖ P ‖ & = & \sqrt{3} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \pm 0 \end{array}

Degree 3, ⟨J ∐ 1⟩ (M curve) 🟢

\begin{array}{rcl} P & = & z (- 3 x^{2} - 3 y^{2} + z^{2}) \\ = & \frac{5 z^{3}}{3} - \frac{{(- \sqrt{3} x + z)}^{3}}{6} - \frac{{(\sqrt{3} x + z)}^{3}}{6} - \frac{{(- \sqrt{3} y + z)}^{3}}{6} - \frac{{(\sqrt{3} y + z)}^{3}}{6} \\ ‖ P ‖ & = & \sqrt{7} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \pm 0 \end{array}

Degree 4, ⟨4⟩ (M curve) 🟢

\begin{array}{rcl} P & = & (- x + y + z) (x - y + z) (x + y - z) (6 x + 6 y + 6 z) - {(x^{2} + y^{2} + z^{2})}^{2} \\ = & - \frac{13 {(x - y)}^{4}}{3} - \frac{13 {(x + y)}^{4}}{3} - \frac{13 {(x - z)}^{4}}{3} - \frac{13 {(x + z)}^{4}}{3} - \frac{13 {(y - z)}^{4}}{3} - \frac{13 {(y + z)}^{4}}{3} + \frac{31 {(x - y - z)}^{4}}{12} + \frac{31 {(x - y + z)}^{4}}{12} + \frac{31 {(x + y - z)}^{4}}{12} + \frac{31 {(x + y + z)}^{4}}{12} \\ ‖ P ‖ & = & \sqrt{197} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \pm 0 \end{array}

Degree 4, ⟨3⟩ (M-1 curve) 🟢

\begin{array}{rcl} P & = & (- x + y + z) (x - y + z) (x + y - z) (2 x + 2 y + 2 z) + {(x^{2} + y^{2} + z^{2})}^{2} \\ = & - 3 x^{4} - 3 y^{4} - 3 z^{4} + \frac{{(x - y)}^{4}}{2} + \frac{{(x + y)}^{4}}{2} + \frac{{(x - z)}^{4}}{2} + \frac{{(x + z)}^{4}}{2} + \frac{{(y - z)}^{4}}{2} + \frac{{(y + z)}^{4}}{2} \\ ‖ P ‖ & = & \sqrt{21} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \pm 0 \end{array}

Degree 5, ⟨𝐽 ∐ 6⟩ (Harnack's M-curve) 🟢

\begin{array}{rcl} P & = & - 0.00106873415699291 x^{5} + 0.0298190870015163 x^{4} y + 0.0298190870015163 x^{4} z - 0.0965192797742039 x^{3} y^{2} + 0.53131622091545 x^{3} y z - 0.0965192797742039 x^{3} z^{2} - 0.0965192797742039 x^{2} y^{3} - 3.08382139247772 x^{2} y^{2} z - 3.08382139247772 x^{2} y z^{2} - 0.0965192797742039 x^{2} z^{3} + 0.0298190870015163 x y^{4} + 0.53131622091545 x y^{3} z - 3.08382139247772 x y^{2} z^{2} + 0.53131622091545 x y z^{3} + 0.0298190870015163 x z^{4} - 0.00106873415699291 y^{5} + 0.0298190870015163 y^{4} z - 0.0965192797742039 y^{3} z^{2} - 0.0965192797742039 y^{2} z^{3} + 0.0298190870015163 y z^{4} - 0.00106873415699291 z^{5} \\ ≃ & 19.0067688827383566386 {(- x + 0.060149842239871802922 y - 0.403906402377617521231 z)}^{5} - 4.10506254927064870458 {(- x + 0.151988408912245202572 y - z)}^{5} + 19.0067688827383566386 {(- 0.403906402377617521231 x - y + 0.060149842239871802922 z)}^{5} + 19.0067688827383566386 {(- 0.403906402377617521231 x + 0.060149842239871802922 y - z)}^{5} + 4.10506254927064870458 {(- 0.151988408912245202572 x + y + z)}^{5} - 19.0067688827383566386 {(- 0.060149842239871802922 x + 0.403906402377617521231 y + z)}^{5} - 19.0067688827383566386 {(- 0.060149842239871802922 x + y + 0.403906402377617521231 z)}^{5} - 11.2705241276892411097 {(- 0.0407005400461146206818 x - 0.0407005400461146206818 y + z)}^{5} + 11.2705241276892411097 {(0.0407005400461146206818 x - y + 0.0407005400461146206818 z)}^{5} + 41.481624760977906961 {(0.0740590551607518339134 x + 0.0740590551607518339134 y + z)}^{5} + 41.481624760977906961 {(0.0740590551607518339134 x + y + 0.0740590551607518339134 z)}^{5} - 11.2705241276892411097 {(x - 0.0407005400461146206818 y - 0.0407005400461146206818 z)}^{5} + 41.481624760977906961 {(x + 0.0740590551607518339134 y + 0.0740590551607518339134 z)}^{5} - 19.0067688827383566386 {(x + 0.403906402377617521231 y - 0.060149842239871802922 z)}^{5} + 4.10506254927064870458 {(x + y - 0.151988408912245202572 z)}^{5} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 6.59982960231662876817 \cdot 10^{-18} \\ dist (P, Δ) & = & 0.0024915028190241140454 \pm 2.1 \cdot 10^{-18} \end{array}

Degree 5, ⟨𝐽 ∐ 5⟩ (M-1 curve) 🟢

\begin{array}{rcl} P & = & 0.138282576385407 x^{5} + 1.60664479990138 \cdot 10^{-74} x^{4} y + 0.587838979474442 x^{4} z - 1.38282576385407 x^{3} y^{2} - 3.45958159975304 \cdot 10^{-74} x^{3} y z - 2.46107002700762 \cdot 10^{-16} x^{3} z^{2} + 1.47505838426774 \cdot 10^{-74} x^{2} y^{3} + 1.17567795894888 x^{2} y^{2} z - 3.70217191641533 \cdot 10^{-75} x^{2} y z^{2} - 1.09348571906146 x^{2} z^{3} + 0.691412881927033 x y^{4} - 2.47113823740793 \cdot 10^{-74} x y^{3} z - 2.46107002700755 \cdot 10^{-16} x y^{2} z^{2} + 2.7937426435764 \cdot 10^{-74} x y z^{3} + 2.09231774897847 \cdot 10^{-16} x z^{4} - 4.55162698598173 \cdot 10^{-75} y^{5} + 0.587838979474442 y^{4} z + 2.22806310010669 \cdot 10^{-75} y^{3} z^{2} - 1.09348571906146 y^{2} z^{3} + 2.12423438761526 \cdot 10^{-75} y z^{4} + 0.520200768721775 z^{5} \\ ≃ & - 0.29605504806173057544 {(- x - 0.72654252800536085922 y - 0.53504137204818599689 z)}^{5} - 0.71454566055256360844 {(- 0.91062288684363371968 x - 3.607937568104080216 \cdot 10^{-75} y - z)}^{5} + 0.62348654107263842188 {(- 0.8606112651663603633 x - 0.62527068422385940514 y - z)}^{5} + 0.62348654107263842188 {(- 0.8606112651663603633 x + 0.62527068422385940514 y - z)}^{5} + 0.71454566055256397067 {(- 0.73670939092327428073 x - 0.53525070328668537012 y + z)}^{5} - 0.66085414860515297247 {(- 0.32491969623290630207 x - y + 0.98842629975056749903 z)}^{5} - 0.66085414860515297247 {(- 0.32491969623290630207 x + y + 0.98842629975056749903 z)}^{5} - 0.71454566055256373042 {(- 0.28139794750145754851 x - 0.86605383042014205249 y - z)}^{5} + 0.71454566055256373042 {(0.28139794750145754851 x - 0.86605383042014205249 y + z)}^{5} - 0.66468499593665377835 {(0.32491969623290631514 x - y - 0.45513337563457402094 z)}^{5} - 0.66468499593665377835 {(0.32491969623290631514 x + y - 0.45513337563457402094 z)}^{5} - 0.71454566055256397067 {(0.73670939092327428073 x - 0.53525070328668537012 y - z)}^{5} + 0.29605504806173057544 {(x - 0.72654252800536085922 y + 0.53504137204818599689 z)}^{5} + 0.84932516656440315741 {(x + 6.8750384056112487783 \cdot 10^{-75} y - 0.94004927325528403985 z)}^{5} - 0.85424854497517219184 {(x + 8.1446624204447317375 \cdot 10^{-75} y - 0.43285756268067151253 z)}^{5} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.2899195370252784891 \cdot 10^{-16} \\ dist (P, Δ) & = & 0.023035530697453850865 \pm 2.0 \cdot 10^{-17} \end{array}

Degree 6

M curve

Degree 6, ⟨9 ∐ 1⟨1⟩⟩ (Harnack's M-curve) 🟢

\begin{array}{rcl} P & = & 9.61142335107702 \cdot 10^{-5} x^{6} + 3.94443008515037 \cdot 10^{-10} x^{5} y + 2.67267233665759 \cdot 10^{-10} x^{5} z + 1.89115375851639 x^{4} y^{2} + 2.58331622515706 x^{4} y z + 0.875510022522388 x^{4} z^{2} + 2.66999394915689 \cdot 10^{-10} x^{3} y^{3} - 6.20517332800208 \cdot 10^{-10} x^{3} y^{2} z - 2.05949268851703 \cdot 10^{-9} x^{3} y z^{2} - 9.31641981873725 \cdot 10^{-10} x^{3} z^{3} - 1.26028860154197 x^{2} y^{4} + 1.7222108157218 x^{2} y^{3} z + 1.75102004537569 x^{2} y^{2} z^{2} - 1.04016485661514 x^{2} y z^{3} - 0.646445147783245 x^{2} z^{4} - 1.32826799302121 \cdot 10^{-10} x y^{5} + 6.55961777153812 \cdot 10^{-10} x y^{4} z - 3.68042052082951 \cdot 10^{-10} x y^{3} z^{2} - 9.32752465572185 \cdot 10^{-10} x y^{2} z^{3} + 3.18455509933182 \cdot 10^{-10} x y z^{4} + 1.98003078320781 \cdot 10^{-10} x z^{5} + 0.210192271668139 y^{6} - 0.861105408365832 y^{5} z + 0.875510021681553 y^{4} z^{2} + 0.346721619732493 y^{3} z^{3} - 0.646445147562636 y^{2} z^{4} - 1.37166228257369 \cdot 10^{-10} y z^{5} + 9.61142335107702 \cdot 10^{-5} z^{6} \\ ≃ & 66.93686669425698903529 {(- x - 0.8218932677946685127487 y + 0.6112970526059105521734 z)}^{6} + 29.83886372938646615504 {(- x - 0.5773502693407670262029 y + 0.7755416011207467205052 z)}^{6} - 161.4834524074586203881 {(- x - 0.5773502693198680564291 y + 0.6680498092570660452873 z)}^{6} + 211.9380952162161431169 {(- x - 0.3755455622667985054818 y + 0.5044618876121104498163 z)}^{6} + 364.9834573493371747307 {(- x - 2.669008966417179101189 \cdot 10^{-13} y + 1.530793450735262160484 \cdot 10^{-10} z)}^{6} - 310.3206708905372230308 {(- x + 0.1794164579134390463943 y - 0.2584626453433536742282 z)}^{6} - 161.483452605925767409 {(- x + 0.5773502692010713005547 y - 0.6680498088140657390602 z)}^{6} + 29.83886377195106663602 {(- x + 0.577350269202969934971 y - 0.7755416006302053481315 z)}^{6} - 67.92753763377424356878 {(- 0.844215781348180053884 x - y + 0.3329343390973398906134 z)}^{6} + 70.72915848987851912085 {(- 1.025471157509130158372 \cdot 10^{-10} x - y - 0.6716387282567188091039 z)}^{6} - 382.7755909860688652925 {(- 8.829686423469717273501 \cdot 10^{-11} x - y - 0.5785481058127065480823 z)}^{6} + 3.183370227680662699357 {(1.530793450735540212018 \cdot 10^{-10} x - 1.060454037763545767665 \cdot 10^{-10} y + z)}^{6} + 290.2376199205781421498 {(0.1658457998032896828955 x - y - 0.478708252551350989984 z)}^{6} + 290.2376199207323084534 {(0.1658457999493018905309 x + y + 0.4787082525005334776673 z)}^{6} - 236.5004226472435980178 {(0.3605824654230248522674 x - y - 0.2704337660992902758391 z)}^{6} - 236.5004226475167270111 {(0.3605824655052172933009 x + y + 0.2704337659888427675485 z)}^{6} + 153.9773960693955920018 {(0.5773502691892671343616 x + y + 1.766848541306840039539 \cdot 10^{-11} z)}^{6} + 153.9773960691108656769 {(0.5773502691899788700535 x - y - 1.94429287584244143221 \cdot 10^{-10} z)}^{6} - 67.92753763359057678674 {(0.8442157814510250374912 x - y + 0.332934338839025927607 z)}^{6} + 66.93686676959816746622 {(x - 0.8218932676399535888585 y + 0.6112970521850770384516 z)}^{6} + 211.9380954128682341349 {(x - 0.3755455622081881381574 y + 0.5044618872279388184746 z)}^{6} - 310.3206707430237639868 {(x + 0.1794164579281873606829 y - 0.2584626456699894184131 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 3.701534591526215018437 \cdot 10^{-16} \\ dist (P, Δ) & = & 0.00009611423351077027294014 \pm 4.4 \cdot 10^{-17} \end{array}

Degree 6, ⟨5 ∐ 1⟨5⟩⟩ (Gudkov's M curve) 🟢

\begin{array}{rcl} P & = & - 0.00132990000822424 y^{6} - 0.0616200461333061 y^{5} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) + 0.00104567643964428 y^{5} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) + 0.0423766780436503 y^{4} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) - 0.00223803203245656265329976974954 y^{4} {(x - z)}^{2} + 0.000244121531585367849579923382741 y^{4} {(x + z)}^{2} + 0.0285392824519227691903999755141 y^{3} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{2} - 0.12925458496204814129448834592 y^{3} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{2} + 0.000389754482938959525133847046016 y^{3} {(x - z)}^{3} + 0.000124842074475882852056131479617 y^{3} {(x + z)}^{3} + 0.0124919547764453464109096998163 y^{2} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{3} - 0.00354416617249992576772138111049 y^{2} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{3} + 0.00000242636930582478404659543119726 y^{2} {(x - z)}^{4} + 0.394230632890865184680961874619 y^{2} {(x - z)}^{2} {(x + z)}^{2} + 0.00000440611823967971157599027992591 y^{2} {(x + z)}^{4} + 0.000427153819177914190023670482077 y (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{4} + 0.000189092943319493398627445523985 y (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{4} - 0.000000247042452224817412433703003204 y {(x - z)}^{5} - 0.0165983876059155323147168928308 y {(x - z)}^{3} {(x + z)}^{2} + 0.172673791645410065331194659611 y {(x - z)}^{2} {(x + z)}^{3} - 0.0000000363005780728571624794169484397 y {(x + z)}^{5} - 0.00000166047194284043445679732716461 (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{5} + 0.0000225364436888175191844757124224 (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{5} - 0.0000000161812343280988403633336293434 {(x - z)}^{6} - 0.00515590206634466020202411939977 {(x - z)}^{4} {(x + z)}^{2} + 0.546067270146320171697595924343 {(x - z)}^{3} {(x + z)}^{3} + 0.00518676318548150563914056476733 {(x - z)}^{2} {(x + z)}^{4} + 6.84766574026006639940924253017 \cdot 10^{-11} {(x + z)}^{6} \\ ≃ & - 167462140.321176701583652489132 {(- x - 0.0841817828027667998013392961308 y + 0.997485287857214101376916696201 z)}^{6} + 55055739.9501972194156296177312 {(- x + 0.114364481107367551732842941469 y - 0.996212862730053857147131461678 z)}^{6} - 8280847.7360865321364889243944 {(- 0.981637388246154630186503873753 x - 0.608825284266300590089525078996 y - z)}^{6} - 70142245.4512914168820429031835 {(- 0.979707711316297963576049017957 x - 0.253739208032426041478352424778 y + z)}^{6} + 1862150.55171690285025281278593 {(- 0.97900135498655133776371225856 x - 0.696336260628075677058575824432 y - z)}^{6} + 38643974.7174452970042259593978 {(- 0.959486200702161240283936700576 x - 0.351627944323446397268378043049 y + z)}^{6} - 16838923.7252090821141439953013 {(- 0.937663269997385300680537406413 x - 0.437809661375089231717735299376 y + z)}^{6} + 4147402.57405932621001633301867 {(- 0.920482049980398491352772998286 x - 0.497097510591201133054486160271 y + z)}^{6} + 21347925.5652220917814409591214 {(0.985538655028765658353322541713 x + 0.48252183183334366208273910617 y + z)}^{6} - 43159828.9194993573114315403831 {(0.990012953172638185336852294883 x + 0.337758659905169118211280524351 y + z)}^{6} + 111921653.018760414725990086373 {(0.994346300208977238119674610775 x + 0.160049406947371112625592154293 y - z)}^{6} + 74322732.0203905853617009877828 {(0.99438795537665266811902468899 x + 0.193089178153952863624172832055 y + z)}^{6} - 113859450.916217918569333495353 {(0.998242726986911135550978449745 x + 0.0641542699349463333893876386532 y + z)}^{6} - 24090665.934844113584111080236 {(x - 0.146840460784293085682587852068 y + 0.992610038479574302394103475322 z)}^{6} + 97398696.9951930851082333271693 {(x - 0.127664630869451728516467679197 y + 0.994391918381327075311969256048 z)}^{6} - 223256453.985527753571557141228 {(x - 0.0988875479900851276194391345382 y + 0.996313881033065652189556837229 z)}^{6} + 162547806.184808119971082792091 {(x - 0.0361740380194784057704085566612 y + 0.998602810810778317617163788778 z)}^{6} + 26165484.2858941698483452614031 {(x - 0.00254821474123598904155948467865 y - 0.99464857619524777621020267677 z)}^{6} - 105380059.313954612248478721823 {(x + 0.0130316203394396028769926166129 y - 0.994109953477023694216423270649 z)}^{6} - 59617163.0160084518612508197032 {(x + 0.0242030510843510414704556666324 y - 0.993232004268181654090580090746 z)}^{6} + 238909354.144697984766328907855 {(x + 0.0358420066614339162037169348755 y - 0.994282639841816167146782195051 z)}^{6} \\ ‖ P ‖ & = & 1.00000000000000015344142921732 \\ ‖ P - P_{1} ‖ & = & 1.56879735402602020685295966417 \cdot 10^{-16} \\ dist (P, Δ) & = & 7.38323798474677818443979735078 \cdot 10^{-11} \pm 8.0 \cdot 10^{-21} \end{array}

Degree 6, ⟨1 ∐ 1⟨9⟩⟩ (Hilbert's M curve) 🟢

\begin{array}{rcl} P & = & - 0.463676008413115 x^{6} + 1.26390330377364 \cdot 10^{-8} x^{5} y - 4.63257086820353 \cdot 10^{-7} x^{5} z - 0.650659383368547 x^{4} y^{2} + 0.262615746075259 x^{4} y z + 1.46209055277665 x^{4} z^{2} - 8.65065554707418 \cdot 10^{-9} x^{3} y^{3} - 4.63703003390089 \cdot 10^{-7} x^{3} y^{2} z + 1.38303903248024 \cdot 10^{-7} x^{3} y z^{2} + 9.72574226759599 \cdot 10^{-7} x^{3} z^{3} - 0.117427822291915 x^{2} y^{4} + 0.704813684540312 x^{2} y^{3} z + 1.66715542428863 x^{2} y^{2} z^{2} - 0.445952509119675 x^{2} y z^{3} - 1.52545873396788 x^{2} z^{4} - 1.60444576368352 \cdot 10^{-8} x y^{5} - 7.21090619761633 \cdot 10^{-8} x y^{4} z + 2.24607818138239 \cdot 10^{-7} x y^{3} z^{2} + 5.80365292711414 \cdot 10^{-7} x y^{2} z^{3} - 1.2530094535893 \cdot 10^{-7} x y z^{4} - 5.06804485221378 \cdot 10^{-7} x z^{5} + 0.0855830564465105 y^{6} + 0.405460169459421 y^{5} z + 0.407757382820572 y^{4} z^{2} - 0.600021803746839 y^{3} z^{3} - 0.994311051098061 y^{2} z^{4} + 0.186036994408672 y z^{5} + 0.527208683439629 z^{6} \\ ≃ & 1850555.33531871767872576778 {(- 0.987851014885905556112132088 x - 0.106139848843755440937596419 y + z)}^{6} - 2374262.60452713702992949187 {(- 0.980532959723679492697858044 x - 0.260438506345591706450492045 y + z)}^{6} + 574945.970085544882459069942 {(- 0.977062516154462552470284354 x + 0.30143858863936072045498432 y - z)}^{6} + 1019052.94417849084735622131 {(- 0.967090225033569233549743316 x + 0.327324104803203692374806041 y - z)}^{6} - 1494724.01187274258520413641 {(- 0.956483498679631273516771421 x - 0.132187301948753472369365336 y - z)}^{6} - 251102.793085081304267999897 {(- 0.954196444521114759678632058 x + 0.368561915919026973492572624 y - z)}^{6} + 1250246.14653578283951981878 {(- 0.848183630165365173823307262 x + 0.411728060938907547077948515 y + z)}^{6} - 1080725.46149209752927984505 {(- 0.643485826161730270419067755 x + 0.675414091581846399692258685 y + z)}^{6} - 1080726.14787067089074780741 {(- 0.643485624785716935678759652 x - 0.675413990329890005506787027 y - z)}^{6} + 976119.653393515255634480123 {(- 0.349228101934146883553936377 x - 0.865220308762646925545633402 y - z)}^{6} - 940224.535310894976997853716 {(0.0000000606365882411177047991768419 x - 0.934658609081793445505550325 y - z)}^{6} - 5257.9847457654571954799101 {(0.0000000686626322520361584348302571 x + y - 0.553685855372668448014504131 z)}^{6} + 5465.53453057965344065469568 {(0.0000000690898645192630566318318138 x + y - 0.558880262824078914522898743 z)}^{6} + 976119.316942738973013719322 {(0.349228246480645719652314676 x - 0.865220374617217823073663835 y - z)}^{6} + 1250247.19316969994434407125 {(0.848183366367290820176171714 x + 0.411727964268361509828039234 y + z)}^{6} - 251102.556602864720486955712 {(0.954196775835579215055329061 x + 0.36856192964203889893423908 y - z)}^{6} - 1494722.60080573586348237709 {(0.956483807555469958027849788 x - 0.132187366980091253029061461 y - z)}^{6} + 1019051.97149199073769497642 {(0.967090558515994396613033063 x + 0.327324112151395466322696239 y - z)}^{6} + 574945.415640468883848205288 {(0.977062851629009230847908183 x + 0.301438591902903056612236266 y - z)}^{6} - 2374264.90226438229688327137 {(0.980532625027766375790344971 x - 0.260438509683555456262138873 y + z)}^{6} + 1850557.13959460227201631055 {(0.987850684956087150871777874 x - 0.106139877279932824579427874 y + z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 6.98840116117683748992644576 \cdot 10^{-17} \\ dist (P, Δ) & = & 5.82146442783478375870245282 \cdot 10^{-9} \pm 2.6 \cdot 10^{-17} \end{array}

M-1 curve

Degree 6, ⟨10⟩ V1 (LM-1 curve) 🟢

This is a M-1 curve, but locally extremal. It is in the same rigid isotopy class than the next one (see the next one for more details).

\begin{array}{rcl} P & = & - {(x^{2} + y^{2} + z^{2})}^{3} + \frac{2 (- x (φ + 2) + z (\sqrt{5} + 3 + \frac{1}{φ})) (x (φ + 2) + z (\sqrt{5} + 3 + \frac{1}{φ})) (x (\sqrt{5} + 3 + \frac{1}{φ}) - y (φ + 2)) (x (\sqrt{5} + 3 + \frac{1}{φ}) + y (φ + 2)) (y (\sqrt{5} + 3 + \frac{1}{φ}) - z (φ + 2)) (y (\sqrt{5} + 3 + \frac{1}{φ}) + z (φ + 2))}{{(- \frac{\sqrt{3} (φ + 2)}{3} + \frac{\sqrt{3} (\sqrt{5} + 3 + \frac{1}{φ})}{3})}^{3} {(\frac{\sqrt{3} (φ + 2)}{3} + \frac{\sqrt{3} (\sqrt{5} + 3 + \frac{1}{φ})}{3})}^{3}} \\ = & \frac{127 {(- φ x + \frac{z}{φ})}^{6}}{30} + \frac{127 {(φ x + \frac{z}{φ})}^{6}}{30} + \frac{127 {(- φ y + \frac{x}{φ})}^{6}}{30} + \frac{127 {(φ y + \frac{x}{φ})}^{6}}{30} + \frac{127 {(- φ z + \frac{y}{φ})}^{6}}{30} + \frac{127 {(φ z + \frac{y}{φ})}^{6}}{30} + \frac{127 {(x - y - z)}^{6}}{30} + \frac{127 {(x - y + z)}^{6}}{30} + \frac{127 {(x + y - z)}^{6}}{30} + \frac{127 {(x + y + z)}^{6}}{30} - (\frac{219}{20} - \frac{73 \sqrt{5}}{15}) (x^{6} {(1 + \sqrt{5})}^{6} + y^{6} {(1 + \sqrt{5})}^{6} + z^{6} {(1 + \sqrt{5})}^{6} + {(- φ x - y (φ + 1) + z)}^{6} + {(- φ x + y (φ + 1) + z)}^{6} + {(φ x - y (φ + 1) + z)}^{6} + {(φ x + y (φ + 1) + z)}^{6} + {(- φ y + x - z (φ + 1))}^{6} + {(- φ y + x + z (φ + 1))}^{6} + {(φ y + x - z (φ + 1))}^{6} + {(φ y + x + z (φ + 1))}^{6} + {(- φ z - x (φ + 1) + y)}^{6} + {(- φ z + x (φ + 1) + y)}^{6} + {(φ z - x (φ + 1) + y)}^{6} + {(φ z + x (φ + 1) + y)}^{6}) \\ ‖ P ‖ & = & \sqrt{2311} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \pm 0 \end{array}

Degree 6, ⟨10⟩ V2 (LM-1 curve) 🟢

This is a M-1 curve, but locally extremal. It is in the same rigid isotopy class than the previous one. It is probably not a local maxima of the distance to the discriminant, because a small random pertubation leads the optimisation process to the previous curve. A path where the 4 critical points inside the triangles are not moving is possible. Here is a video of the transition:

\begin{array}{rcl} P & = & 1458 (- \frac{4 x^{2}}{9} - \frac{4 y^{2}}{9} + \frac{5 z^{2}}{9}) (- \frac{4 x^{2}}{9} + \frac{5 y^{2}}{9} - \frac{4 z^{2}}{9}) (\frac{5 x^{2}}{9} - \frac{4 y^{2}}{9} - \frac{4 z^{2}}{9}) + {(x^{2} + y^{2} + z^{2})}^{3} \\ = & - \frac{9236 {(\frac{\sqrt{2} x}{2} - \frac{\sqrt{2} y}{2})}^{6}}{3} - \frac{9236 {(\frac{\sqrt{2} x}{2} + \frac{\sqrt{2} y}{2})}^{6}}{3} - \frac{9236 {(\frac{\sqrt{2} x}{2} - \frac{\sqrt{2} z}{2})}^{6}}{3} - \frac{9236 {(\frac{\sqrt{2} x}{2} + \frac{\sqrt{2} z}{2})}^{6}}{3} - \frac{9236 {(\frac{\sqrt{2} y}{2} - \frac{\sqrt{2} z}{2})}^{6}}{3} - \frac{9236 {(\frac{\sqrt{2} y}{2} + \frac{\sqrt{2} z}{2})}^{6}}{3} + \frac{15255 {(\frac{x}{3} - \frac{2 y}{3} - \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{x}{3} - \frac{2 y}{3} + \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{x}{3} + \frac{2 y}{3} - \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{x}{3} + \frac{2 y}{3} + \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} - \frac{2 y}{3} - \frac{z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} - \frac{2 y}{3} + \frac{z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} - \frac{y}{3} - \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} - \frac{y}{3} + \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} + \frac{y}{3} - \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} + \frac{y}{3} + \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} + \frac{2 y}{3} - \frac{z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} + \frac{2 y}{3} + \frac{z}{3})}^{6}}{4} - 6744 {(\frac{\sqrt{3} x}{3} - \frac{\sqrt{3} y}{3} - \frac{\sqrt{3} z}{3})}^{6} - 6744 {(\frac{\sqrt{3} x}{3} - \frac{\sqrt{3} y}{3} + \frac{\sqrt{3} z}{3})}^{6} - 6744 {(\frac{\sqrt{3} x}{3} + \frac{\sqrt{3} y}{3} - \frac{\sqrt{3} z}{3})}^{6} - 6744 {(\frac{\sqrt{3} x}{3} + \frac{\sqrt{3} y}{3} + \frac{\sqrt{3} z}{3})}^{6} \\ ‖ P ‖ & = & \sqrt{91213} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \pm 0 \end{array}

Degree 6, ⟨8 ∐ 1⟨1⟩⟩ ( M-1 curve) 🟢

\begin{array}{rcl} P & = & - 0.000106347928803112 x^{6} + 0.0254351539332643 x^{5} y + 0.0140123171058973 x^{5} z + 1.86633907684415 x^{4} y^{2} + 2.39459691774544 x^{4} y z + 0.781339376354797 x^{4} z^{2} + 0.0162196871056121 x^{3} y^{3} - 0.481304454833255 x^{3} y^{2} z - 0.128745899946072 x^{3} y z^{2} + 0.16086792752555 x^{3} z^{3} - 1.22431337100098 x^{2} y^{4} + 1.90072675265439 x^{2} y^{3} z + 1.86339176459758 x^{2} y^{2} z^{2} - 0.905123855944642 x^{2} y z^{3} - 0.638406023963338 x^{2} z^{4} - 0.00905330090553513 x y^{5} + 0.183693651882951 x y^{4} z - 0.389919067353814 x y^{3} z^{2} + 0.160867524385695 x y^{2} z^{3} - 0.0539878330316615 x y z^{4} - 0.305774248490535 x z^{5} + 0.200956514088202 y^{6} - 0.885439723544508 y^{5} z + 0.930664332902469 y^{4} z^{2} + 0.425951644026074 y^{3} z^{3} - 0.60703517629456 y^{2} z^{4} - 0.176575020499063 y z^{5} - 0.0583581799181141 z^{6} \\ ≃ & - 13.49427104058058400279 {(- x - 0.8864734619601845292679 y + 0.7488679138564983081543 z)}^{6} + 3.773434414522096225715 {(- x - 0.4341110379859453838344 y + 0.9757186983822281284177 z)}^{6} - 56.00395911781829076896 {(- x - 0.3335352834482230659119 y + 0.5907539734656744261243 z)}^{6} - 87.13047451903487283607 {(- x + 0.4903198306921383510721 y - 0.5640890893962866821744 z)}^{6} + 33.41670029617187317796 {(- x + 0.7594635966847908299074 y - 0.5492826772403722717766 z)}^{6} - 25.47512631689678129823 {(- 0.8710707915004480898342 x + y - 0.32185438504325247025 z)}^{6} + 62.20912096869476213731 {(- 0.5773371095808022246397 x + y + 0.0000482675153022197407249 z)}^{6} - 32.74650874704860611611 {(- 0.57137509561169846768 x - 0.3299243137022147262858 y + z)}^{6} + 113.1731947340172407319 {(- 0.2698019358150217516801 x - y - 0.3549976166502567471852 z)}^{6} + 124.8783985615405750237 {(- 0.1265670827923300938173 x + y + 0.4410134085517008923327 z)}^{6} + 29.920180187023168659 {(- 0.05998285817681178358633 x - y - 0.6235056244070192504964 z)}^{6} - 163.979582671023285515 {(0.06787252995202668553274 x + y + 0.5077351831183688470966 z)}^{6} - 99.22894131122969056842 {(0.3431659460313159320406 x - y - 0.2566763743318841259878 z)}^{6} + 21.08214136454185010653 {(0.3819394187779207980435 x + 0.2205560675875185904179 y - z)}^{6} - 69.02347714283232975826 {(0.5015239759088414793044 x + y + 0.08853359583251880433371 z)}^{6} + 27.80242157984568822544 {(0.76850760832297883072 x + y - 0.2832020556073931499606 z)}^{6} + 3.255793631938826434988 {(0.8977305915882028707153 x + 0.6651816129989601698294 y - z)}^{6} + 15.48500267557354486693 {(x - 0.5001106248538337174235 y + 0.6957723622959460195584 z)}^{6} + 113.8102315950355714082 {(x - 0.2661410325480309665194 y + 0.3546045983507451986781 z)}^{6} - 133.9138371082498468374 {(x - 0.0588257069952612443029 y + 0.0792290167206917781562 z)}^{6} + 106.1956712493591847924 {(x + 0.1324056831200719436415 y - 0.2265448171445526397571 z)}^{6} + 21.54461172743200138537 {(x + 0.5773793626114741467479 y - 0.8753927026771764098507 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 9.843902382321646750079 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.000292319017125182931889 \pm 4.6 \cdot 10^{-17} \end{array}

Degree 6, ⟨5 ∐ 1⟨4⟩⟩ (M-1 curve) 🟢

\begin{array}{rcl} P & = & - 0.0116095915095109 y^{6} - 0.0218349191022255 y^{5} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) + 0.262021012623916 y^{5} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) - 0.640030727882121 y^{4} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) + 0.228317001353184306777777123898 y^{4} {(x - z)}^{2} - 0.287326162550341102086548517036 y^{4} {(x + z)}^{2} + 0.474457898523348575725577802587 y^{3} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{2} - 0.16287326072949959154811949702 y^{3} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{2} - 0.318938274231448723219062902338 y^{3} {(x - z)}^{3} - 0.214128176667383344695114078597 y^{3} {(x + z)}^{3} - 0.287877793226545596214945418345 y^{2} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{3} + 0.12996131001541402374907991765 y^{2} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{3} + 0.00095574150254287663919239292909 y^{2} {(x - z)}^{4} - 0.341704375640636826005902548786 y^{2} {(x - z)}^{2} {(x + z)}^{2} + 0.355356220489455920663601773413 y^{2} {(x + z)}^{4} + 0.209473267546922359283456671619 y (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{4} - 0.0248656229816706687840710543469 y (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{4} + 0.0476837545485906587067951512813 y {(x - z)}^{5} - 0.193615106314295143153490900016 y {(x - z)}^{3} {(x + z)}^{2} + 0.0583671584980486517846979558651 y {(x - z)}^{2} {(x + z)}^{3} - 0.0572560734797863349641612352968 y {(x + z)}^{5} - 0.0155510005098031616863752376756 (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{5} - 0.00979724506106207007791055596086 (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{5} + 0.0630566117569658773911456250971 {(x - z)}^{6} - 0.177466271583034684855562090888 {(x - z)}^{4} {(x + z)}^{2} + 0.0170353267992079983328412851051 {(x - z)}^{3} {(x + z)}^{3} + 0.15797880011669801492146802957 {(x - z)}^{2} {(x + z)}^{4} - 0.043480439811159385388172893272 {(x + z)}^{6} \\ ≃ & 3029.05796445619340208013157484 {(- x - 0.237085230038854841213800806609 y - 0.147048017072966490539576761739 z)}^{6} + 28807.3868089462858835402240536 {(- x - 0.0353924164496377861920924541438 y - 0.0698076846905757575242939223155 z)}^{6} + 76692.8777977406120652758089295 {(- x + 0.156417919064914279592255088498 y - 0.0262631703536262017348042726158 z)}^{6} - 38823.3965289419453338842310933 {(- x + 0.218993477282843444183354935137 y - 0.0395375396801437497744556086214 z)}^{6} + 15433.1881224692964932342293084 {(- x + 0.282762446301578387294113459388 y - 0.0664981234186415992396215978495 z)}^{6} - 3524.72513269380896721155585974 {(- x + 0.331015860780695811796690870161 y - 0.094056853912076907194872903089 z)}^{6} - 7552.44984336411499938436152773 {(- 0.248754596845103367812087299681 x - 0.777415596270804515563454455642 y + z)}^{6} - 37872.7996045856675267842959559 {(- 0.0789970905505745991257561769915 x - 0.158130712545122523212095492515 y + z)}^{6} + 26798.1713635653524390835930818 {(- 0.00957022252839425818626577491572 x - 0.257203892692341401288884511546 y - z)}^{6} - 66308.4752135414341645273269108 {(- 0.00775187619126081546562283195729 x - 0.190038248538256121387950339659 y - z)}^{6} - 6230.65480864369476412213609515 {(- 0.0023208871314435694859075668058 x - 0.302531636637744240284690651573 y - z)}^{6} + 15772.7690224216233643718700708 {(0.0232687097873560290649216527018 x + 0.226835170595233229407216312969 y + z)}^{6} + 53460.9932168317030796451193032 {(0.0241895232674251996431071923563 x - 0.0506009013953990779255872921439 y - z)}^{6} + 20296.2749600253469272499274624 {(0.153232537522690683612309197637 x + 0.428595222201734021905600385839 y - z)}^{6} + 4654.81292815964647103738 {(0.27520171035908223056663478212 x + y - 0.505135269374291384946115140328 z)}^{6} - 14728.8164897060611328661 {(0.286944475809479266946850005529 x + y - 0.57890612027651467910410141844 z)}^{6} + 4584.19620546970663556069 {(0.288426263758134217988762254075 x + y - 0.506021889187568217501630455468 z)}^{6} + 7029.75819838094277327652 {(0.300050838735315221340496313686 x + y - 0.813421263582712094510010972703 z)}^{6} - 18729.8784247754192476928507358 {(x - 0.1663120810506045931370813421 y + 0.0138984359502482859078617064659 z)}^{6} - 50283.1184993329729669191591134 {(x - 0.0737354862953895856751101087098 y + 0.0393448183306186763747991016544 z)}^{6} - 12601.4266838305248651769661287 {(x + 0.14833055327769509509528641532 y + 0.110431815308108436778705628755 z)}^{6} \\ ‖ P ‖ & = & 1.00000000000000005374383481261 \\ ‖ P - P_{1} ‖ & = & 8.61150001610586011346072823592 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.00000148930335241601295817243077877 \pm 2.6 \cdot 10^{-17} \end{array}

Degree 6, ⟨4 ∐ 1⟨5⟩⟩ (M-1 curve) 🔴

\begin{array}{rcl} P & = & 0.54545792261561 x^{6} + 0.163478488475309 x^{5} y - 0.019960905743059 x^{5} z + 0.376152642688739 x^{4} y^{2} - 0.181759629577558 x^{4} y z - 1.63632758519634 x^{4} z^{2} - 0.0981961246892561 x^{3} y^{3} - 0.0729695797078391 x^{3} y^{2} z - 0.324522334015374 x^{3} y z^{2} + 0.0399353430133078 x^{3} z^{3} + 0.000478660666325454 x^{2} y^{4} - 0.22413649345713 x^{2} y^{3} z - 0.766243512794597 x^{2} y^{2} z^{2} + 0.359050782559892 x^{2} y z^{3} + 1.63645915117095 x^{2} z^{4} - 0.0665796116970206 x y^{5} - 0.0488513800080948 x y^{4} z + 0.0961456180429968 x y^{3} z^{2} + 0.0728102830398332 x y^{2} z^{3} + 0.161067582296572 x y z^{4} - 0.0199747348677223 x z^{5} - 0.00552008870228949 y^{6} - 0.0719627935779878 y^{5} z - 0.0395215576061384 y^{4} z^{2} + 0.21925999604176 y^{3} z^{3} + 0.390528682567556 y^{2} z^{4} - 0.177294209697579 y z^{5} - 0.545589614341794 z^{6} \\ ≃ & 158472.65873753654664603427 {(- x - 0.027322160123195567913518614 y - 0.99914965506842697353872344 z)}^{6} - 40373.326610418192161757399 {(- x + 0.0097946521820098429250011883 y - 0.99150755003883086933419391 z)}^{6} - 70136.668463826096477663969 {(- x + 0.043966035910689921490189858 y - 0.99764402266782067406568179 z)}^{6} + 82696.030196455869904697854 {(- x + 0.12205297633407684065208657 y + 0.98989827955107926384848236 z)}^{6} + 7348.3641125227619533378533 {(- x + 0.36891304587635463527947784 y + 0.96981123133940867319631555 z)}^{6} + 17307.505323796639700568634 {(- 0.99843529578688813700649476 x + 0.092444996366039772389985775 y - z)}^{6} + 37015.162348294725930761034 {(- 0.99378507693433360879185062 x + 0.026626094747297697070213592 y + z)}^{6} - 51530.534299459849920324789 {(- 0.96953193999545607610792276 x - 0.38740910639181663706014891 y + z)}^{6} + 22566.728701026919707404781 {(- 0.94889685149583297530549615 x - 0.62542130573684235952460756 y + z)}^{6} - 53519.791294578721715730247 {(- 0.80640816218665352871182494 x - 0.6690472769941880354774501 y - z)}^{6} + 30625.147426939265142185503 {(- 0.63906460387777082519219623 x - 0.95149891673507744999030355 y - z)}^{6} - 47454.926257237805915583397 {(- 0.34849213030269753527735392 x - y - 0.82977459023746870467320942 z)}^{6} - 17848.445564610997355036923 {(0.018455935308515100628740675 x - y - 0.67869296081093559371435121 z)}^{6} + 47058.188462645045376128878 {(0.12731637697584991834475963 x + y + 0.72097618113586223784079692 z)}^{6} + 1475.1408904416739304848009 {(0.88466308494418709629797531 x + y - 0.96634095863928884263325078 z)}^{6} + 82694.036231943522486205544 {(0.91625411290235961747312553 x + 0.40016936292835381438564457 y + z)}^{6} - 7119.0750764271004982923467 {(0.92953245654586689995183537 x + 0.84976669624995661625202936 y - z)}^{6} - 116661.30123813329585306773 {(0.97676583740482090810185137 x + 0.17590881359190208732899691 y + z)}^{6} + 96272.463411474106490187309 {(0.98646601152133238662809893 x + 0.1665357714865653521964444 y - z)}^{6} - 153032.73034883579473632571 {(0.99731818302281644703920066 x + 0.00054101407794298766551083892 y - z)}^{6} - 33624.467783464144516230948 {(x - 0.25307203399730905118618923 y - 0.97716258877766731525524394 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 0.86296246035754733338819352 \\ dist (P, Δ) & = & 3.6400494955296569522542647 \cdot 10^{-8} \pm 1.2 \cdot 10^{-8} \end{array}

Degree 6, ⟨1 ∐ 1⟨8⟩⟩ (M-1 curve) 🟢

\begin{array}{rcl} P & = & - 0.451903628853404 x^{6} - 0.0717803922844663 x^{5} y - 0.00812414035275845 x^{5} z - 0.62317275203232 x^{4} y^{2} + 0.253319343875893 x^{4} y z + 1.43608528361424 x^{4} z^{2} + 0.0812729711766909 x^{3} y^{3} + 0.125060194951806 x^{3} y^{2} z + 0.172076678093925 x^{3} y z^{2} + 0.0169603754082284 x^{3} z^{3} - 0.0665401738246307 x^{2} y^{4} + 0.771513176864937 x^{2} y^{3} z + 1.66356036360845 x^{2} y^{2} z^{2} - 0.424872175364114 x^{2} y z^{3} - 1.50981522890084 x^{2} z^{4} + 0.120806410504159 x y^{5} + 0.173049443630852 x y^{4} z - 0.00642790777049121 x y^{3} z^{2} - 0.11052226083349 x y^{2} z^{3} - 0.0989607840904805 x y z^{4} - 0.00880453657357438 x z^{5} + 0.10015178739329 y^{6} + 0.461003706359938 y^{5} z + 0.429174731476599 y^{4} z^{2} - 0.640329382914588 y^{3} z^{3} - 1.01498038036713 y^{2} z^{4} + 0.174109666770399 y z^{5} + 0.525753067906681 z^{6} \\ ≃ & - 106306.1887362124873957456 {(- 0.9993634715333990078787353 x + 0.153872906924252608076343 y - z)}^{6} + 46447.79657751104575545918 {(- 0.9892845274318638371578355 x + 0.2504430255599882101560317 y - z)}^{6} + 50861.52341402067423479685 {(- 0.936273883532047088463444 x - 0.411926914777282443262249 y + z)}^{6} - 57137.31059377921510481008 {(- 0.9026423229038353601371416 x + 0.3524761041055239178071634 y + z)}^{6} - 20159.88391794757361346885 {(- 0.8769593317582649533043935 x - 0.5560219809693024292098175 y + z)}^{6} - 71148.0907022833257535742 {(- 0.8646567672384502707787668 x - 0.3213037901558632109932939 y - z)}^{6} + 4609.227628524211451261722 {(- 0.8169624737172663469454926 x - 0.6809747357254479001325849 y + z)}^{6} + 61895.14874447926377475223 {(- 0.6579719593780318313657338 x - 0.5982790527308145318740144 y - z)}^{6} - 45843.6358491864602386378 {(- 0.3832019567895911762920409 x + 0.8601284116075163273101252 y + z)}^{6} - 54108.41654382338419343393 {(- 0.3572404015392006331740369 x - 0.8130520751654102790124379 y - z)}^{6} - 4814.026801483622478191963 {(- 0.3299641409740540275685339 x - y + 0.6914455803647159051567848 z)}^{6} + 48184.0218777878356233919 {(0.007887421449770037028026598 x - 0.9124768080730048675711533 y - z)}^{6} + 10594.47077878512446522783 {(0.3854694617461868762488194 x + y - 0.74408397266476988882435 z)}^{6} - 6076.59591079425522713224 {(0.4699396483781391937912555 x + y - 0.8165839249659527339415492 z)}^{6} + 48520.14527411211527949401 {(0.6990496639065036148441194 x - 0.6571304398927882584639205 y - z)}^{6} + 84399.40450772960281653395 {(0.9707337693333885079688205 x + 0.04681382457615768100344562 y + z)}^{6} - 11768.95717732661715372987 {(0.9710934877016440769175349 x - 0.3136521641039347059557624 y + z)}^{6} + 24406.24671787848729089922 {(0.9721518178939867145715083 x + 0.3310389691809187544718598 y - z)}^{6} - 100839.1254809333348908093 {(0.9778709924756455368680412 x + 0.2502707365984675826840698 y - z)}^{6} + 73157.60440559560741478987 {(0.9842311689671504037147632 x - 0.02239614757487132022972356 y - z)}^{6} + 26456.50503643707390581128 {(x - 0.2071633648704274556581009 y + 0.9929741940626885464425663 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 3.908341361448111370051955 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.0000001414243685904218718519495 \pm 8.9 \cdot 10^{-18} \end{array}

Degree 6, ⟨1⟨9⟩⟩ (M-1 curve) 🟢

\begin{array}{rcl} P & = & - 0.129323311431944 x^{6} - 9.77698378875362 \cdot 10^{-8} x^{5} y - 1.99884974675323 \cdot 10^{-7} x^{5} z + 0.449203673504653 x^{4} y^{2} + 1.66162636325709 x^{4} y z + 1.06046630798817 x^{4} z^{2} - 2.40367636372275 \cdot 10^{-7} x^{3} y^{3} - 2.0879938953301 \cdot 10^{-7} x^{3} y^{2} z + 9.4822240982206 \cdot 10^{-7} x^{3} y z^{2} + 7.5746718893791 \cdot 10^{-7} x^{3} z^{3} - 0.813614302232198 x^{2} y^{4} + 1.4816635888513 x^{2} y^{3} z + 1.85238667179824 x^{2} y^{2} z^{2} - 1.66994424179561 x^{2} y z^{3} - 1.35548616753138 x^{2} z^{4} + 5.83986169350786 \cdot 10^{-8} x y^{5} - 4.59465733327726 \cdot 10^{-7} x y^{4} z + 3.34374739495702 \cdot 10^{-7} x y^{3} z^{2} + 8.29710547372079 \cdot 10^{-7} x y^{2} z^{3} - 4.1069667658711 \cdot 10^{-7} x y z^{4} - 4.50573559255312 \cdot 10^{-7} x z^{5} - 0.0717495331608978 y^{6} - 0.368640323687943 y^{5} z + 1.29101550789213 y^{4} z^{2} - 0.226295953949779 y^{3} z^{3} - 1.40139503226646 y^{2} z^{4} + 0.323320011346218 y z^{5} + 0.474532234970588 z^{6} \\ ≃ & 22.02777655082157034004 {(- 0.9821583431501009595829 x + 0.5116710396361840877116 y - z)}^{6} + 103.2800437849274639437 {(- 0.90642081834147760262 x - 0.2129231089522908645794 y + z)}^{6} + 103.2801363817382376951 {(- 0.9064205080763653363298 x + 0.2129231195704157733669 y - z)}^{6} - 130.8484250365963050296 {(- 0.8874087273135969652847 x - 0.4568806091738105354284 y + z)}^{6} - 102.5951342335314198207 {(- 0.823399840503016690868 x + 0.1094613762420094247671 y + z)}^{6} - 102.5952177913842255302 {(- 0.8233995690053324453845 x - 0.1094613228358291359125 y - z)}^{6} + 98.55028221376500543891 {(- 0.6732120569538844363557 x + 0.4455697398320619802936 y + z)}^{6} + 98.55034783727493238521 {(- 0.6732118382458660171822 x - 0.4455696588653720730251 y - z)}^{6} + 117.3080421638847303245 {(- 0.6715677245578332846855 x - 0.5816643866312538762921 y + z)}^{6} + 117.3081200871753491278 {(- 0.6715674581238964549445 x + 0.5816643536747666032379 y - z)}^{6} - 82.5459802492201174383 {(- 0.4745776542640880420145 x - 0.7692719345491675393053 y - z)}^{6} - 133.2541175622320877002 {(- 0.3501178241726073896596 x - 0.6529411134336771244727 y + z)}^{6} - 133.2541637093330882325 {(- 0.3501176085431287597773 x + 0.6529410921380494133687 y - z)}^{6} + 87.82403018071747768549 {(- 0.2264828608107651121269 x - y - 0.9588004034945348499595 z)}^{6} + 144.2173196939497055458 {(- 9.821388531798327061482 \cdot 10^{-8} x - 0.6744391523762243417404 y + z)}^{6} + 61.5479848411392417005 {(3.940730687801850541922 \cdot 10^{-8} x - y - 0.7620707965684137644444 z)}^{6} - 205.0330110289547264426 {(4.646918553613794120501 \cdot 10^{-8} x - y - 0.8477453547513293367957 z)}^{6} + 87.82403576785777399076 {(0.2264829696555769473945 x - y - 0.9588003559919450903118 z)}^{6} - 82.54594150090596429423 {(0.4745778202328122728589 x - 0.7692720169513903563875 y - z)}^{6} - 130.84853988948742636 {(0.8874084112497379924049 x - 0.456880583880077011673 y + z)}^{6} + 22.02775515141835108968 {(0.9821586909813924599345 x + 0.511671076501819646715 y - z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 8.32967190528990188686 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.00008204983450139948663002 \pm 4.4 \cdot 10^{-17} \end{array}

M-2 curve

Degree 6, ⟨9⟩ V1 (M-2 curve) 🟢

\begin{array}{rcl} P & = & 0.233594218911071 x^{6} - 4.07281985498405 \cdot 10^{-75} x^{5} y - 0.641371605237325 x^{5} z - 1.19756444601588 x^{4} y^{2} - 3.28365793954453 \cdot 10^{-75} x^{4} y z - 0.222372275773307 x^{4} z^{2} + 8.83887233794854 \cdot 10^{-75} x^{3} y^{3} + 1.28181304653891 x^{3} y^{2} z - 1.48771813088546 \cdot 10^{-74} x^{3} y z^{2} + 1.26885941607468 x^{3} z^{3} + 1.9507985788156 x^{2} y^{4} + 2.24449448634003 \cdot 10^{-74} x^{2} y^{3} z - 0.36871912291207 x^{2} y^{2} z^{2} - 2.58989821746096 \cdot 10^{-75} x^{2} y z^{3} + 0.551484339071583 x^{2} z^{4} + 6.62333018549243 \cdot 10^{-75} x y^{5} + 1.94937187910436 x y^{4} z + 6.04912344403714 \cdot 10^{-75} x y^{3} z^{2} - 3.8376791129805 x y^{2} z^{3} + 1.79639523084266 \cdot 10^{-74} x y z^{4} + 0.00248338516152038 x z^{5} + 0.0219277217380189 y^{6} + 9.12354016960645 \cdot 10^{-75} y^{5} z - 0.199177739908626 y^{4} z^{2} - 1.32081247263847 \cdot 10^{-74} y^{3} z^{3} + 0.531736987010761 y^{2} z^{4} - 7.23810912421521 \cdot 10^{-75} y z^{5} + 0.0219309176572576 z^{6} \\ ≃ & 1.3326266660341046323 {(- x + 3.6430498147724724272 \cdot 10^{-75} y - 0.43795358974010924024 z)}^{6} - 1.261257992632033923 {(- x + 0.39940246393036384518 y - 0.41675809870776277002 z)}^{6} + 0.21072843337813558989 {(- x + 0.44988507701728562628 y + 0.86523189687794641166 z)}^{6} - 0.46733676143531410792 {(- x + 0.78732336377197753388 y + 0.48612223128711203534 z)}^{6} - 0.36091896015695601885 {(- 0.78725097500634446248 x + 3.266752678644165356 \cdot 10^{-75} y + z)}^{6} - 0.36755184859019422208 {(- 0.38856251729211339466 x + 0.67748104852777372621 y - z)}^{6} - 1.8575902831142389878 {(- 0.14598238505029940927 x + y + 0.3879163839197476543 z)}^{6} + 0.35152954404501042255 {(- 0.09827785809561970797 x - y - 0.79712320854716740372 z)}^{6} + 0.58885873182829753959 {(- 0.0025826529276869108346 x + 7.6540289020792697481 \cdot 10^{-75} y + z)}^{6} + 2.1078646314140450322 {(- 3.6729025277459629179 \cdot 10^{-75} x + 1.0 y + 3.7056169973791166748 \cdot 10^{-75} z)}^{6} + 0.35152954404501042255 {(0.09827785809561970797 x - y + 0.79712320854716740372 z)}^{6} - 1.8575902831142389878 {(0.14598238505029940927 x + y - 0.3879163839197476543 z)}^{6} - 0.36755184859019422208 {(0.38856251729211339466 x + 0.67748104852777372621 y + z)}^{6} + 0.56601673248372274211 {(0.57864523166166288986 x - y - 0.50066231140999931746 z)}^{6} + 0.56601673248372274211 {(0.57864523166166288986 x + y - 0.50066231140999931746 z)}^{6} + 0.10210732575725475781 {(x - 0.72117989556373802765 y + 0.98178241664420310171 z)}^{6} + 0.88923759735830286289 {(x - 0.57735219467731735925 y + 0.0025826529276869108346 z)}^{6} - 1.261257992632033923 {(x + 0.39940246393036384518 y + 0.41675809870776277002 z)}^{6} + 0.21072843337813558989 {(x + 0.44988507701728562628 y - 0.86523189687794641166 z)}^{6} + 0.88923759735830286289 {(x + 0.57735219467731735925 y + 0.0025826529276869108346 z)}^{6} + 0.10210732575725475781 {(x + 0.72117989556373802765 y + 0.98178241664420310171 z)}^{6} - 0.46733676143531410792 {(x + 0.78732336377197753388 y - 0.48612223128711203534 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 5.5927272433871342368 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.021927721738018910195 \pm 2.1 \cdot 10^{-17} \end{array}

Degree 6, ⟨9⟩ V2 (M-2 curve) 🟢

\begin{array}{rcl} P & = & 0.0219495555284178 x^{6} + 1.18288037331849 \cdot 10^{-75} x^{5} y + 2.73892026746226 \cdot 10^{-75} x^{5} z + 2.44425828411031 x^{4} y^{2} + 2.35584925300597 \cdot 10^{-75} x^{4} y z - 0.567379925676993 x^{4} z^{2} + 4.01100165909071 \cdot 10^{-75} x^{3} y^{3} - 6.79368469367151 \cdot 10^{-75} x^{3} y^{2} z - 8.07589687114957 \cdot 10^{-75} x^{3} y z^{2} - 6.75045244218622 \cdot 10^{-75} x^{3} z^{3} - 0.730516640154519 x^{2} y^{4} - 2.27774056547665 \cdot 10^{-76} x^{2} y^{3} z - 4.541921534574 x^{2} y^{2} z^{2} - 3.02925231471749 \cdot 10^{-76} x^{2} y z^{3} + 1.36660405165139 x^{2} z^{4} - 8.73803397298739 \cdot 10^{-76} x y^{5} - 8.67429783101763 \cdot 10^{-76} x y^{4} z - 7.21022361106744 \cdot 10^{-76} x y^{3} z^{2} + 8.2999359937226 \cdot 10^{-75} x y^{2} z^{3} - 1.14388882458369 \cdot 10^{-75} x y z^{4} - 9.96830048655807 \cdot 10^{-76} x z^{5} + 0.0219495555284178 y^{6} + 1.64720084051449 \cdot 10^{-75} y^{5} z + 1.67438043505708 y^{4} z^{2} - 3.80400590177022 \cdot 10^{-75} y^{3} z^{3} - 0.197522112270671 y^{2} z^{4} + 2.5347243732675 \cdot 10^{-76} y z^{5} - 0.0219495555284178 z^{6} \\ ≃ & 0.48552648016210895885 {(- x - 0.36305989413510690028 y - 0.70362513445393674551 z)}^{6} - 2.0738422672660902304 {(- x + 2.0207709650940687257 \cdot 10^{-76} y - 0.41840210183238467397 z)}^{6} - 0.1232935227599853887 {(- 0.90455831430929836001 x - 0.90746535231012908448 y - z)}^{6} - 0.1232935227599853887 {(- 0.90455831430929836001 x + 0.90746535231012908448 y - z)}^{6} - 0.1232935227599853887 {(- 0.90455831430929836001 x + 0.90746535231012908448 y + z)}^{6} + 0.52998133387842742121 {(- 0.58362576032920407855 x - y - 0.41065375408241728212 z)}^{6} - 2.0454318873726962324 {(- 0.36323773261648906968 x + y - 9.9230492400645168175 \cdot 10^{-76} z)}^{6} + 2.2447175562590688288 {(- 5.486197037362095218 \cdot 10^{-76} x + 1.0 y - 5.120199901551843588 \cdot 10^{-76} z)}^{6} - 0.76620182664182756604 {(3.479451837973786882 \cdot 10^{-76} x + 9.6250400377735845124 \cdot 10^{-76} y + 1.0 z)}^{6} + 0.25345817771451703142 {(0.35138005651326891305 x - 0.51598482822372065326 y - z)}^{6} + 0.25345817771451703142 {(0.35138005651326891305 x - 0.51598482822372065326 y + z)}^{6} + 0.25345817771451703142 {(0.35138005651326891305 x + 0.51598482822372065326 y - z)}^{6} + 0.25345817771451703142 {(0.35138005651326891305 x + 0.51598482822372065326 y + z)}^{6} - 2.0454318873726962324 {(0.36323773261648906968 x + y - 4.5006764214256752722 \cdot 10^{-76} z)}^{6} + 0.52998133387842742121 {(0.58362576032920407855 x - y - 0.41065375408241728212 z)}^{6} + 0.52998133387842742121 {(0.58362576032920407855 x - y + 0.41065375408241728212 z)}^{6} + 0.52998133387842742121 {(0.58362576032920407855 x + y - 0.41065375408241728212 z)}^{6} - 0.1232935227599853887 {(0.90455831430929836001 x + 0.90746535231012908448 y - z)}^{6} + 2.4213979974502550851 {(1.0 x - 2.4867044864824053357 \cdot 10^{-76} y + 2.1626631369228810049 \cdot 10^{-75} z)}^{6} + 0.48552648016210895885 {(x - 0.36305989413510690028 y - 0.70362513445393674551 z)}^{6} + 0.48552648016210895885 {(x - 0.36305989413510690028 y + 0.70362513445393674551 z)}^{6} - 2.0738422672660902304 {(x + 3.5544614993863025597 \cdot 10^{-76} y - 0.41840210183238467397 z)}^{6} + 0.48552648016210895885 {(x + 0.36305989413510690028 y - 0.70362513445393674551 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 5.4414327558519677865 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.021949555528417773575 \pm 1.4 \cdot 10^{-17} \end{array}

Degree 6, ⟨7 ∐ 1⟨1⟩⟩ (M-2 curve) 🟢

\begin{array}{rcl} P & = & 0.000132672198832339 x^{6} - 0.0543572490064376 x^{5} y - 0.0249424663708054 x^{5} z + 1.74066511326169 x^{4} y^{2} + 2.41188905106373 x^{4} y z + 0.858042219742042 x^{4} z^{2} - 0.0385570035451854 x^{3} y^{3} + 0.471544890171046 x^{3} y^{2} z + 0.0733108573545426 x^{3} y z^{2} - 0.389363179158887 x^{3} z^{3} - 1.16615517063327 x^{2} y^{4} + 1.7975755696013 x^{2} y^{3} z + 2.2669111327925 x^{2} y^{2} z^{2} - 0.343293300246056 x^{2} y z^{3} - 0.327052492280756 x^{2} z^{4} + 0.0167039363675761 x y^{5} - 0.20764715188056 x y^{4} z + 0.628412704109518 x y^{3} z^{2} - 0.319642170369938 x y^{2} z^{3} - 0.957667847413321 x y z^{4} + 5.15204517300554 \cdot 10^{-5} x z^{5} + 0.193910481306253 y^{6} - 0.889157479708478 y^{5} z + 0.94832183965086 y^{4} z^{2} + 0.53160541877523 y^{3} z^{3} - 0.851456912821865 y^{2} z^{4} - 0.0286304994898774 y z^{5} + 0.000326335790990764 z^{6} \\ ≃ & - 15.3061499398211369525 {(- x - 0.672222721846934317712 y + 0.537516500852503568431 z)}^{6} + 28.654305140528710109 {(- x + 0.103305722380559994006 y - 0.17855893817209177136 z)}^{6} + 1.53005267731985965641 {(- x + 0.337827879701964292644 y - 0.903168306607951039997 z)}^{6} - 3.29874752455404608783 {(- x + 0.947956843327229127369 y - 0.699532685214376336888 z)}^{6} + 6.64168612511549996669 {(- 0.937306825631873831163 x - y + 0.344566132179619606379 z)}^{6} + 8.53392913307421061201 {(- 0.719784494175152116276 x + y - 0.207522227146428745776 z)}^{6} - 19.0862318673025793946 {(- 0.576831326489994862816 x - y - 0.0125684341206988868819 z)}^{6} + 32.8456027593273758274 {(- 0.300661684388587589874 x - y - 0.286178500802042883638 z)}^{6} - 10.2369473346028325368 {(- 0.124870286314069340634 x + 0.059460658773031182144 y + z)}^{6} + 7.23858515185959082412 {(- 0.0671172523410846499908 x - y - 0.658632298429197517308 z)}^{6} - 43.870892105122365759 {(0.0631380228664598185104 x + y + 0.470654007180725524699 z)}^{6} + 20.9595229963731411679 {(0.0829127787449897991273 x - 0.0603951222671428042158 y + z)}^{6} + 29.9355295952208363615 {(0.183677912977542889454 x - y - 0.382578307705548297695 z)}^{6} - 13.135676484223285382 {(0.35248566258411719127 x - 0.215893206437799956743 y + z)}^{6} - 18.9762116372251321502 {(0.447573595218274210056 x - y - 0.149338741943815175809 z)}^{6} + 0.804749745152765293031 {(0.871848465261995155248 x - 0.794130869924125923461 y + z)}^{6} + 6.02600016532837144248 {(x - 0.586356189255020881713 y + 0.75784005179607644116 z)}^{6} - 16.9492736186416009964 {(x - 0.307759744809354796258 y + 0.537728083079189565594 z)}^{6} - 32.0486371550217434626 {(x + 0.106227154346600171687 y - 0.123995884570961822202 z)}^{6} + 23.6777239244394534209 {(x + 0.362632078490122482405 y - 0.380200418617900758479 z)}^{6} + 2.52642158572538857086 {(x + 0.680510284042484352565 y - 0.7614045650513667284 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 5.26117191444276960979 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.0012559623808573861052 \pm 2.2 \cdot 10^{-17} \end{array}

Degree 6, ⟨6 ∐ 1⟨2⟩⟩ (EM-2 curve) 🟢

\begin{array}{rcl} P & = & 0.00589370417177452 x^{6} - 0.00810722420831899 x^{5} y - 0.10447541397529 x^{5} z - 0.13246170003432 x^{4} y^{2} + 0.507503425686876 x^{4} y z + 0.48496566263209 x^{4} z^{2} - 0.340529737547038 x^{3} y^{3} + 1.06649896955336 x^{3} y^{2} z - 4.25866713675959 x^{3} y z^{2} - 0.102094774223009 x^{3} z^{3} - 0.387024873194602 x^{2} y^{4} + 0.82973870683771 x^{2} y^{3} z + 6.87333566012994 x^{2} y^{2} z^{2} + 0.500807048068058 x^{2} y z^{3} - 0.021632585240089 x^{2} z^{4} - 0.198554452044143 x y^{5} + 1.15205593386011 x y^{4} z + 1.04043383730238 x y^{3} z^{2} + 0.961863155460903 x y^{2} z^{3} + 0.157899579839491 x y z^{4} + 0.000589006101698443 x z^{5} - 0.0260996888277111 y^{6} + 0.200758307624054 y^{5} z - 0.0915693420332 y^{4} z^{2} + 0.169052103282536 y^{3} z^{3} + 0.0328301791661001 y^{2} z^{4} + 0.000389369942468445 y z^{5} - 0.000247824170496405 z^{6} \\ ≃ & 87.66646864564133967407 {(- x - 0.1507293597227276952201 y - 0.1157839790016475252289 z)}^{6} + 265.6957559909633296016 {(- x + 0.065650241518947071828 y - 0.0859866326669048964606 z)}^{6} - 153.7999933798103593101 {(- x + 0.3279036352493385149154 y - 0.03357610317218884702966 z)}^{6} + 56.51911811773487533917 {(- x + 0.7048539152754089039994 y + 0.002037252838309579270864 z)}^{6} - 50.83277840913819544554 {(- 0.7755717129177099727248 x + y + 0.04441005916920915745317 z)}^{6} - 124.6938202985261106282 {(- 0.2149458091187146144034 x - 0.3078411813492471006455 y + z)}^{6} + 83.66722354267773963712 {(- 0.1954058467210668898676 x + y + 0.1191397881834638853994 z)}^{6} + 127.02202342069828286 {(- 0.1951164913941355558577 x + y + 0.4034946388763416214009 z)}^{6} - 107.2131512627209183429 {(- 0.1472917695691492199013 x - 0.02571699358497896138242 y + z)}^{6} + 22.36546402489446321938 {(- 0.1292507806402819707198 x + 0.671894154008624656853 y + z)}^{6} + 406.7913764011825171958 {(- 0.09415465369625705064322 x - 0.1364706990341131093822 y + z)}^{6} - 121.2378219049532003115 {(- 0.0008313540981224980538622 x - 0.004069789319835573878709 y + z)}^{6} - 106.9836117662367414535 {(0.02802591735275618142236 x - 0.1493366314922560001229 y + z)}^{6} - 28.15423506287319001111 {(0.1751266439942546998774 x - y - 0.7963553267184028747649 z)}^{6} - 301.3127160309538666799 {(0.2514791432050271865961 x - y - 0.1823871641707819966009 z)}^{6} + 22.00298558166379189693 {(0.3299484396081005613924 x + 0.4709992882946424591097 y - z)}^{6} + 160.6497480531579803449 {(0.4325505542742250263193 x - y - 0.09660458117281437004966 z)}^{6} + 22.14541879693895914916 {(0.6800810289776988237387 x + 0.1023913279096786582978 y + z)}^{6} - 353.1992244036750545862 {(x + 0.09638920334209254978348 y + 0.1750846373868174072439 z)}^{6} + 133.6307590682774476811 {(x + 0.1503574351733348949438 y + 0.3976667330107091116902 z)}^{6} - 28.63718263211169353803 {(x + 0.1692261177187930400198 y + 0.7917080778383553338454 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 8.970496853187705608646 \cdot 10^{-16} \\ dist (P, Δ) & = & 0.0002488484400222630224711 \pm 1.4 \cdot 10^{-17} \end{array}

Degree 6, ⟨5 ∐ 1⟨3⟩⟩ (M-2 curve) 🟢

\begin{array}{rcl} P & = & 0.0566058234512678 y^{6} - 0.171961360199937 y^{5} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) - 0.397211461343717 y^{5} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) + 1.50905823298469 y^{4} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) - 0.0515635943132141671974011387647 y^{4} {(x - z)}^{2} + 0.283650919679543034312985128054 y^{4} {(x + z)}^{2} - 1.25821490621297327194838544528 y^{3} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{2} + 0.043804304224213518048625104484 y^{3} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{2} - 0.0045921432210591941795591454895 y^{3} {(x - z)}^{3} + 0.147177214260917839293021927164 y^{3} {(x + z)}^{3} - 0.766854899280219991424667135838 y^{2} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{3} - 0.262634408624459466219111421615 y^{2} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{3} - 0.00806551604164006813446619048591 y^{2} {(x - z)}^{4} + 0.0271469694784216251937092323488 y^{2} {(x - z)}^{2} {(x + z)}^{2} + 0.0395113427447199225261975641388 y^{2} {(x + z)}^{4} - 0.196502370915257879024906628729 y (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{4} + 0.0188703126322104716705574389835 y (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{4} + 0.0000394555932029899482595958654476 y {(x - z)}^{5} + 0.322421868376131173798566036403 y {(x - z)}^{3} {(x + z)}^{2} + 0.588727984137322599180556875715 y {(x - z)}^{2} {(x + z)}^{3} + 0.00650699775107701574271587178976 y {(x + z)}^{5} - 0.0161779796694962246710340461117 (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{5} - 0.0012124570980378873732496344614 (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{5} + 0.00000183300955924030358009551352344 {(x - z)}^{6} - 0.00921120938664906491188943959969 {(x - z)}^{4} {(x + z)}^{2} + 0.348200811201710980213874790934 {(x - z)}^{3} {(x + z)}^{3} + 0.0543892205337456469127488389859 {(x - z)}^{2} {(x + z)}^{4} + 0.000431376468606762414077088196862 {(x + z)}^{6} \\ ≃ & 133.377770406053049782257337049 {(- x - 0.0206842193110781209301659722301 y - 0.873341342827548782981880684325 z)}^{6} - 169.451832024867638416705543805 {(- x + 0.228845240515755622729117726942 y - 0.899666473637724988262440125204 z)}^{6} + 30.8205110660838854592366373442 {(- x + 0.287426466122905572084086799891 y - 0.987664130485750558372400848776 z)}^{6} + 117.849348786663746171023563199 {(- x + 0.36729215459737265724859503157 y - 0.763410112195440064434464884106 z)}^{6} - 80.2967509339445014991751998112 {(- x + 0.460179781663349138982550438295 y - 0.55856766259204457683761882812 z)}^{6} - 97.1683295717388883521360127905 {(- 0.898109762346899768947109079593 x + 0.333760628262793961335321773249 y + z)}^{6} - 68.4497259099545883094016523164 {(- 0.896398477839011595502847509451 x - 0.183243551402464405307416516489 y + z)}^{6} - 71.68890124817261756693 {(- 0.872398009726647177114601675413 x - y - 0.37583475667165223846584387914 z)}^{6} - 11.7687940144534109378620877814 {(- 0.820217712180373633939555762246 x + 0.880053692112848394247990228462 y + z)}^{6} + 21.1819639847296038900855649086 {(- 0.817186078248456960323699544087 x - 0.342039913758981716553528154103 y + z)}^{6} + 135.7062325764763375589 {(0.272924381404767285798118231816 x + y + 0.129130459751116832559229838159 z)}^{6} - 393.5066141651327067569 {(0.412111357130633209212303977554 x + y + 0.0684081365161938505920405815313 z)}^{6} + 103.6998975554917837758 {(0.413414128773520236055090532917 x + y - 0.00682003303047670774276933763058 z)}^{6} + 220.6644066596932444003 {(0.588984018570354170271684872065 x + y + 0.147859868219218316960576388365 z)}^{6} - 17.2864636109094634573724462237 {(0.706146082619513891615722747466 x - 0.807176310853451958723790084557 y - z)}^{6} + 69.8738741068317305458214029436 {(0.810662666331774627883468122642 x - 0.668213758365625664320350333108 y - z)}^{6} + 127.191129512722060036658810187 {(0.957506372494049781466268137984 x - 0.0254841004837921523220900239809 y - z)}^{6} + 30.4135120817987756306514456259 {(x - 0.5049566784008382105448406598 y + 0.37690673457562699745583735981 z)}^{6} - 32.7875071331339610621035044559 {(x + 0.0458143759667160575886278358751 y - 0.949627748886465023221920240889 z)}^{6} - 107.96154484404736125555642488 {(x + 0.34753396829313138486562781783 y + 0.769093284339501345634278966817 z)}^{6} + 70.2369346135104134516661291822 {(x + 0.716197409643470909131809815113 y + 0.615274970803313338383936549758 z)}^{6} \\ ‖ P ‖ & = & 0.999999999999999997860503337354 \\ ‖ P - P_{1} ‖ & = & 8.77105056398522528764082333413 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.00010567637298890290819206198019 \pm 1.7 \cdot 10^{-17} \end{array}

Degree 6, ⟨4 ∐ 1⟨4⟩⟩ (M-2 curve) 🟢

\begin{array}{rcl} P & = & 0.0159616683326227 y^{6} + 0.0962949830807944 y^{5} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) - 0.133400310739169 y^{5} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) + 1.00849648913753 y^{4} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) - 0.456369829976032870266777763391 y^{4} {(x - z)}^{2} + 0.0149326405367441600313682670276 y^{4} {(x + z)}^{2} - 0.472167365591932897128657486974 y^{3} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{2} - 0.389332491274201331066251441371 y^{3} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{2} - 0.200851398466210284048099172856 y^{3} {(x - z)}^{3} + 0.0589099147689395163899314573208 y^{3} {(x + z)}^{3} - 0.449625806667668857704831698941 y^{2} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{3} - 0.503827333274907509787349490514 y^{2} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{3} - 0.0414218109837022410713025522 y^{2} {(x - z)}^{4} + 0.862837111243200172161493810564 y^{2} {(x - z)}^{2} {(x + z)}^{2} + 0.0221556392391357125093431790219 y^{2} {(x + z)}^{4} - 0.103511320211923468770365275304 y (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{4} - 0.0921960878112641490922385401063 y (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{4} - 0.00407074222928948897636357597291 y {(x - z)}^{5} + 0.0732161912921543295075870494032 y {(x - z)}^{3} {(x + z)}^{2} + 0.157285084817273259449628794376 y {(x - z)}^{2} {(x + z)}^{3} + 0.0029786240676828409241447546426 y {(x + z)}^{5} - 0.00712742194819554348964012703579 (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{5} - 0.00504011542763791852003218584403 (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{5} - 0.000150250112016921420548740129242 {(x - z)}^{6} + 0.00807529113530183056957056919601 {(x - z)}^{4} {(x + z)}^{2} + 0.44627839697788440620485062027 {(x - z)}^{3} {(x + z)}^{3} + 0.0277357172745772082023130167272 {(x - z)}^{2} {(x + z)}^{4} + 0.000131148287325940043727967587728 {(x + z)}^{6} \\ ≃ & 132.169492336133669220140950317 {(- x - 0.982661721834351532237900447207 y - 0.6660590650545652041227747719 z)}^{6} - 277.332422327685590694679561789 {(- x - 0.537635551257128498736410970049 y - 0.763146684331858330489761542244 z)}^{6} - 402.804854268417820189288154142 {(- x + 0.192411263675551282059744848449 y - 0.979321017187724898847887887419 z)}^{6} - 117.946669318940019869890937225 {(- x + 0.198695482017446243680129655349 y + 0.982159598655553993989651035118 z)}^{6} + 176.423964883682897075041065378 {(- 0.988704101840649984710586051136 x + 0.431356470108409294248761769065 y - z)}^{6} - 62.256179301099812268809441381 {(- 0.96265217390238033814313295779 x + 0.662705907190005250136013442159 y - z)}^{6} + 536.891501979612830727122315605 {(- 0.950596663302417138820465441456 x + 0.164189701489158676466151019374 y + z)}^{6} + 15.3964466948914514924050833125 {(- 0.87651720080166402363923206489 x + 0.848600417794706547795977924972 y - z)}^{6} + 152.520395134668907929664804077 {(- 0.795126875681533597874427563605 x + 0.454744311494436950583797886337 y + z)}^{6} + 450.575501308754468365731966412 {(- 0.691431786693491417425173958792 x - 0.308596493173031470096268580608 y + z)}^{6} + 181.33907817774036413227 {(- 0.606942327982968716314148787114 x - y - 0.306029435190276438228507718586 z)}^{6} + 190.02821180790225959345 {(- 0.569116757565007842085822151832 x - y - 0.337130434621228494722986231441 z)}^{6} - 375.64097382833627511133683011 {(- 0.466134654998773049627212337988 x - 0.546845750810251047115838266171 y + z)}^{6} + 240.694949723759244358585750512 {(- 0.208571566837983642461227379409 x - 0.737048205335014724260943101639 y + z)}^{6} - 89.092481164925407553719749525 {(- 0.0385695312009894269796167273849 x + 0.848117929693814694657353830072 y - z)}^{6} - 45.6622163818367999460448371377 {(0.686313805223677818239485903959 x - 0.541258522021584791608578478253 y - z)}^{6} - 480.97451229382199366493 {(0.687383503665503756289460897853 x + y + 0.401159363545639571152210462398 z)}^{6} - 475.715546586857390110907738531 {(0.862312957649694115726365324868 x + 0.0534620692447174510464008090213 y - z)}^{6} - 296.802397564824554701092853571 {(0.903702873445308041101099350774 x - 0.319215694851418090410982365221 y - z)}^{6} + 110.895875370405220052875360992 {(0.934147431819097588385047747903 x - 0.284636824864688982333754859528 y + z)}^{6} + 387.602380481101308718653097711 {(x + 0.147154269025771807477121029 y + 0.869418240278985373310916973063 z)}^{6} \\ ‖ P ‖ & = & 1.00000000000000000834219939156 \\ ‖ P - P_{1} ‖ & = & 3.28840331857390369712284516297 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.0000321839633299168736662881776962 \pm 5.1 \cdot 10^{-18} \end{array}

Degree 6, ⟨3 ∐ 1⟨5⟩⟩ (M-2 curve) 🔴

It seems there is no exceptional polynomial in the rigid isotopy class of this polynomial. The paper only proves there existence for locally extremal curve and it is possible that when we maximise the distance to the discriminant, quasi-cusp point are mandatory. This could be the only case for M-2 sextic. We will investigate more this case.

\begin{array}{rcl} P & = & 0.544912811279858 x^{6} + 0.163184071318975 x^{5} y - 0.0180502721412947 x^{5} z + 0.379688270707065 x^{4} y^{2} - 0.173522117321053 x^{4} y z - 1.63883427401915 x^{4} z^{2} - 0.0977872493151784 x^{3} y^{3} - 0.078678993674308 x^{3} y^{2} z - 0.325487126505175 x^{3} y z^{2} + 0.0398359555542617 x^{3} z^{3} + 0.00636312920319746 x^{2} y^{4} - 0.231755793359361 x^{2} y^{3} z - 0.761329100292231 x^{2} y^{2} z^{2} + 0.351936505276977 x^{2} y z^{3} + 1.63929977845366 x^{2} z^{4} - 0.0661355889330344 x y^{5} - 0.0555856028297564 x y^{4} z + 0.102407266819225 x y^{3} z^{2} + 0.0804625534286655 x y^{2} z^{3} + 0.161835061720841 x y z^{4} - 0.0219246747565946 x z^{5} - 0.0049509157250979 y^{6} - 0.0701866516177861 y^{5} z - 0.046321262534803 y^{4} z^{2} + 0.230687331066375 y^{3} z^{3} + 0.380411931746573 y^{2} z^{4} - 0.177931509239157 y z^{5} - 0.545240702063767 z^{6} \\ ≃ & z^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.75797650841174 \\ dist (P, Δ) & = & 0.545240702063767 \pm 0 \end{array}

Degree 6, ⟨2 ∐ 1⟨6⟩⟩ (EM-2 curve) 🟢

\begin{array}{rcl} P & = & - 0.396663870421844 x^{6} + 0.21640136821547 x^{5} y - 0.0109419425863422 x^{5} z - 0.817302526479225 x^{4} y^{2} + 0.48072714803358 x^{4} y z + 1.27206642925114 x^{4} z^{2} + 0.0508999223953115 x^{3} y^{3} - 0.175349776021804 x^{3} y^{2} z - 0.451149966359061 x^{3} y z^{2} + 0.0232569761630139 x^{3} z^{3} - 0.163029808841459 x^{2} y^{4} + 1.04638338228826 x^{2} y^{3} z + 1.51195475956509 x^{2} y^{2} z^{2} - 1.04470250583701 x^{2} y z^{3} - 1.35783369451186 x^{2} z^{4} - 0.234906246896496 x y^{5} - 0.166627696213582 x y^{4} z + 0.0221136877114014 x y^{3} z^{2} + 0.181835573825987 x y^{2} z^{3} + 0.235225289182257 x y z^{4} - 0.0123739559359567 x z^{5} + 0.194998705706446 y^{6} + 0.566460762192207 y^{5} z - 0.00753890289944141 y^{4} z^{2} - 1.13007440315571 y^{3} z^{3} - 0.670730783712748 y^{2} z^{4} + 0.566435465154493 y z^{5} + 0.482446648620434 z^{6} \\ ≃ & - 17235.78104699555602219795 {(- x - 0.2851841792469923933438584 y - 0.9466525988899291541909117 z)}^{6} + 5899.186159204494470457514 {(- x - 0.1537138956699653039203101 y - 0.9312014362520819345449942 z)}^{6} + 11386.00571526472876238833 {(- x + 0.06854216572955960610609681 y + 0.9683731905951197557098248 z)}^{6} + 15282.44910133099647675234 {(- 0.9548768722027620234105237 x + 0.3723098942956839721710419 y + z)}^{6} + 8332.793100638581890372477 {(- 0.9524616191849746306972701 x - 0.6088550471512966962292393 y - z)}^{6} - 12283.77805359011180986917 {(- 0.7738609928751712105257009 x + 0.6246203299665761664362382 y + z)}^{6} - 7005.341876831375375881862 {(- 0.6889837205598740218737574 x - 0.8815560402753904459176806 y - z)}^{6} + 17902.62673387365047275938 {(- 0.6691231242756413752775106 x + 0.6996944209207052546993786 y - z)}^{6} + 10250.59896621390251221698 {(- 0.4826816020533281057962529 x + 0.8546009818196989515643475 y + z)}^{6} - 8637.513759238015929132038 {(- 0.1074991467068690581851935 x + y + 0.9985571075818130489368707 z)}^{6} + 8228.001358637972002469773 {(0.3006376799025058256396074 x + y + 0.9806010843155724918476483 z)}^{6} - 1178.711579136890235298786 {(0.4249775146432928589613815 x - y + 0.9217325980972658543063802 z)}^{6} + 3450.569190889195512791331 {(0.5069223619506019457306845 x - 0.977234802278066955217925 y + z)}^{6} - 9242.362451686301584648707 {(0.5827266931496944532825182 x - 0.8368117392225369924233741 y + z)}^{6} - 5161.022078620527266766613 {(0.6585249967681531507989871 x - 0.6822493668311044227544239 y + z)}^{6} - 7176.854033535390256110567 {(0.7710661153949144979264051 x - 0.5732827026749252858151699 y + z)}^{6} - 24264.08468940339337265363 {(x - 0.169966209443957174835989 y - 0.9706045077988007649295833 z)}^{6} + 6651.727835993959867207816 {(x - 0.1176627272258693763146435 y - 0.948508687895651300485655 z)}^{6} - 2784.698770358627717095026 {(x + 0.001988443429673666335425937 y - 0.983369335070545251761446 z)}^{6} + 6426.084880810182898193698 {(x + 0.1137951409827975332447314 y + 0.9509181896888931615652811 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.417451527066488381549453 \cdot 10^{-16} \\ dist (P, Δ) & = & 0.0000006564983740116376063184517 \pm 2.4 \cdot 10^{-17} \end{array}

Degree 6, ⟨1 ∐ 1⟨7⟩⟩ (M-2 curve) 🟢

\begin{array}{rcl} P & = & - 0.41945536802886 x^{6} + 0.028993731909103 x^{5} y - 0.000416427033813448 x^{5} z - 0.73386284514219 x^{4} y^{2} + 0.222401957764597 x^{4} y z + 1.33823230163809 x^{4} z^{2} + 0.0643225011471911 x^{3} y^{3} + 0.0292488850166352 x^{3} y^{2} z - 0.0391105063275275 x^{3} y z^{2} + 0.00164786573131822 x^{3} z^{3} - 0.149642938781006 x^{2} y^{4} + 0.749057358230338 x^{2} y^{3} z + 1.92564385510751 x^{2} y^{2} z^{2} - 0.444089755858393 x^{2} y z^{3} - 1.41948070134616 x^{2} z^{4} + 0.0284897603366637 x y^{5} + 0.0383265617404168 x y^{4} z - 0.0357728193985202 x y^{3} z^{2} - 0.0260295080983391 x y^{2} z^{3} + 0.0104956092378329 x y z^{4} - 0.00125248310804325 x z^{5} + 0.124743311827073 y^{6} + 0.587499460423241 y^{5} z + 0.60016964261906 y^{4} z^{2} - 0.737067023290336 y^{3} z^{3} - 1.18611597297508 y^{2} z^{4} + 0.221272208846738 y z^{5} + 0.50062946591728 z^{6} \\ ≃ & 344.64979776340304420673 {(- x + 0.0065541837735052926745797 y + 0.94852618333004122608239 z)}^{6} + 705.95170097500758849877 {(- 0.96958746701895594088832 x - 0.157104399946823674607 y + z)}^{6} + 907.36394995371805194109 {(- 0.92350565670244617317432 x - 0.27026142675400045094596 y - z)}^{6} + 894.75646796756812165345 {(- 0.91166338934582210407317 x + 0.32214474221903685276918 y + z)}^{6} - 378.74589047422582750588 {(- 0.87470709013776939248997 x - 0.37138017127558289526834 y + z)}^{6} - 383.75396739395549823025 {(- 0.84824222998763690931373 x + 0.41857641990746802524905 y - z)}^{6} + 118.22538754884354569451 {(- 0.73455352692777637665289 x + 0.58466122972542927084377 y - z)}^{6} - 780.96162966970951072471 {(- 0.69068305614221958172142 x + 0.59872230363409437951785 y + z)}^{6} + 747.43084139055606289934 {(- 0.40308016900823241762691 x - 0.77135317988131260749782 y - z)}^{6} + 743.08153944851685862013 {(- 0.36234340534924693718079 x + 0.79346630584328532575705 y + z)}^{6} - 736.14225013740324319833 {(0.022604337798313722604243 x + 0.85588795960811077520876 y + z)}^{6} - 522.51944745883379105868 {(0.028554502701567673911522 x + y - 0.51893736622445798105105 z)}^{6} + 1552.9579004111115887783 {(0.028616741556076179255086 x + y - 0.56791646064084173940255 z)}^{6} - 522.26823782230209349785 {(0.05537715788078541555904 x - y + 0.59684832706829494313602 z)}^{6} - 507.75989966269806769253 {(0.11308295533571381673233 x + y - 0.59944298339340966085746 z)}^{6} - 789.40168149484787174174 {(0.71915454058012621538437 x + 0.55835758320174237237316 y + z)}^{6} + 116.87696574374808259753 {(0.77001201052238961170447 x + 0.54377089590866993663369 y - z)}^{6} + 716.3838621922238555017 {(0.95444269893499279329208 x - 0.21032569112731360777723 y + z)}^{6} - 1217.9234454323899770209 {(x - 0.05463791416367195898829 y - 0.99496193086619797731316 z)}^{6} + 337.26866874284649682724 {(x - 0.049313110979070997132349 y + 0.95450084039245943265957 z)}^{6} - 1210.3500672098986997649 {(x - 0.0011244234106276242662102 y + 0.99853961271826891352886 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 6.5390976660035473780022 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.000011994344962266722172027 \pm 2.8 \cdot 10^{-17} \end{array}

Degree 6, ⟨1⟨8⟩⟩ ND (M-2 curve) 🟢

\begin{array}{rcl} P & = & - 0.0569131186273626 x^{6} + 0.383114047180663 x^{5} y + 0.153243745312552 x^{5} z + 0.337963171029331 x^{4} y^{2} + 1.55128936973495 x^{4} y z + 0.947692334597727 x^{4} z^{2} + 0.617080146495235 x^{3} y^{3} + 0.305792460898885 x^{3} y^{2} z - 0.0963660100845385 x^{3} y z^{2} + 0.114733990923011 x^{3} z^{3} - 0.691432092276297 x^{2} y^{4} + 1.48274307179365 x^{2} y^{3} z + 1.91598867489581 x^{2} y^{2} z^{2} - 1.47497826455634 x^{2} y z^{3} - 1.29304380361951 x^{2} z^{4} + 0.226986206633784 x y^{5} + 0.0523600474657622 x y^{4} z - 0.336421952405771 x y^{3} z^{2} + 0.181683895120387 x y^{2} z^{3} + 0.0354322337637058 x y z^{4} - 0.1808012358826 x z^{5} - 0.186157618896397 y^{6} - 0.146778081524262 y^{5} z + 1.23788184203379 y^{4} z^{2} - 0.198291814226667 y^{3} z^{3} - 1.41961599962061 y^{2} z^{4} + 0.225749802333319 y z^{5} + 0.490326461797867 z^{6} \\ ≃ & 3.023059161086369566253 {(- x - 0.7295900707679609426627 y + 0.8583020692886437514841 z)}^{6} - 10.37117733901341619789 {(- x - 0.5474218590749623998473 y + 0.5551562181660846480915 z)}^{6} + 13.03397972950599445043 {(- 0.834159473385038433711 x + 0.6166744274181943531486 y - z)}^{6} + 63.84664137729121621048 {(- 0.806166031247588753085 x + 0.2768321664588475418478 y - z)}^{6} - 21.4174463071929532043 {(- 0.7895671954665703450162 x - 0.7528832591891313258095 y + z)}^{6} - 35.86917862121763687023 {(- 0.6877686962363412376636 x + 0.5645999769521003491136 y + z)}^{6} + 24.47530304340465525703 {(- 0.1489369523183632452971 x + y + 0.8127856742347556685826 z)}^{6} + 42.04268551700011549117 {(- 0.1227978517991130954246 x - 0.9846102128830464278423 y - z)}^{6} - 55.74109675840533250196 {(- 0.08933889938248413840947 x + 0.7341325702290287599171 y - z)}^{6} - 72.32810848572391917397 {(0.1541179368471773880934 x - y - 0.9285482846935460130833 z)}^{6} + 38.52840981168800055117 {(0.3267004021352321213103 x + 0.7621124167237599661056 y - z)}^{6} - 50.1950919184048093615 {(0.4060083832907184882809 x + 0.7384936155715972724402 y + z)}^{6} + 38.60786393406401361944 {(0.4298075169490289146249 x - 0.8830949205160567712704 y - z)}^{6} + 64.91867580931234275691 {(0.4589568260918594481322 x - 0.6644365789805858774987 y + z)}^{6} + 59.66658114692545400463 {(0.6239021617600478343995 x + 0.4173080678638084320847 y + z)}^{6} - 79.44996107903871834027 {(0.7312424689385785599798 x - 0.5413997793315543584103 y + z)}^{6} - 62.6475352413163233412 {(0.7641297907245391606935 x + 0.06739830768135000099811 y + z)}^{6} + 27.79994824753885761194 {(0.9197444598983802370906 x - 0.1885744503532228507407 y - z)}^{6} - 36.77023416246866655765 {(x + 0.2068376919837120186228 y - 0.886777293986987640034 z)}^{6} + 7.847528165462753317188 {(x + 0.3683230210504675438072 y - 0.7178413613348603339733 z)}^{6} + 26.88114982467846468716 {(x + 0.5290832425378395385011 y - 0.8810148933857361297159 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 6.444597321843617188844 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.0002102182576757141635492 \pm 1.8 \cdot 10^{-17} \end{array}

Degree 6, ⟨1⟨8⟩⟩ D (M-2 curve) 🟢

\begin{array}{rcl} P & = & 0.00047830888439636 x^{6} + 1.97729542651369 \cdot 10^{-86} x^{5} y + 5.15637816550416 \cdot 10^{-87} x^{5} z + 1.34901792671098 x^{4} y^{2} - 3.04861822763262 \cdot 10^{-87} x^{4} y z - 0.927631586332153 x^{4} z^{2} + 3.16441980225004 \cdot 10^{-87} x^{3} y^{3} - 2.6108667695848 \cdot 10^{-87} x^{3} y^{2} z - 1.90677970260907 \cdot 10^{-86} x^{3} y z^{2} - 3.36649545733576 \cdot 10^{-87} x^{3} z^{3} + 1.34901792671098 x^{2} y^{4} - 2.84518320056041 \cdot 10^{-88} x^{2} y^{3} z - 2.98080318905296 x^{2} y^{2} z^{2} + 2.64161178954934 \cdot 10^{-87} x^{2} y z^{3} + 1.40516354200258 x^{2} z^{4} - 2.1478112375648 \cdot 10^{-86} x y^{5} - 5.28655719060703 \cdot 10^{-87} x y^{4} z + 1.56706265296712 \cdot 10^{-86} x y^{3} z^{2} + 5.33419009896911 \cdot 10^{-87} x y^{2} z^{3} + 1.53754976175109 \cdot 10^{-87} x y z^{4} - 3.64608236286858 \cdot 10^{-88} x z^{5} + 0.00047830888439636 y^{6} + 2.68077928199968 \cdot 10^{-87} y^{5} z - 0.927631586332153 y^{4} z^{2} - 1.48106804761482 \cdot 10^{-87} y^{3} z^{3} + 1.40516354200258 y^{2} z^{4} - 4.78392978687652 \cdot 10^{-88} y z^{5} - 0.529745845686385 z^{6} \\ ≃ & 1.589507374275634313154 {(- 1.0 x + 7.336451359318464778957 \cdot 10^{-87} y - 2.775031762222988443084 \cdot 10^{-87} z)}^{6} - 18.14043459667601807177 {(- 0.8277341603719508826234 x - 0.3874078909018352984282 y - z)}^{6} - 18.14043459667601807177 {(- 0.8277341603719508826234 x + 0.3874078909018352984282 y + z)}^{6} - 2.787017736002609434885 {(- 0.8275051790378073440092 x - 0.8275051790378073440092 y - z)}^{6} - 2.787017736002609434885 {(- 0.8275051790378073440092 x + 0.8275051790378073440092 y - z)}^{6} - 2.787017736002609434885 {(- 0.8275051790378073440092 x + 0.8275051790378073440092 y + z)}^{6} + 20.76074458474428818709 {(- 0.7015625904065544847387 x - 0.7015625904065544847387 y + z)}^{6} - 18.14043459667601807177 {(- 0.3874078909018352984282 x - 0.8277341603719508826234 y + z)}^{6} - 18.14043459667601807177 {(- 0.3874078909018352984282 x + 0.8277341603719508826234 y - z)}^{6} + 18.17470588318876109328 {(- 1.994078366799941458147 \cdot 10^{-87} x - 0.8691668928268026710331 y + z)}^{6} + 1.589507374275634313154 {(7.969124118784105641906 \cdot 10^{-87} x + 1.0 y + 1.442727320665996895082 \cdot 10^{-87} z)}^{6} + 18.17470588318876109328 {(9.973265537948382004341 \cdot 10^{-87} x + 0.8691668928268026710331 y + z)}^{6} - 18.14043459667601807177 {(0.3874078909018352984282 x - 0.8277341603719508826234 y - z)}^{6} - 18.14043459667601807177 {(0.3874078909018352984282 x + 0.8277341603719508826234 y + z)}^{6} + 20.76074458474428818709 {(0.7015625904065544847387 x - 0.7015625904065544847387 y - z)}^{6} + 20.76074458474428818709 {(0.7015625904065544847387 x - 0.7015625904065544847387 y + z)}^{6} + 20.76074458474428818709 {(0.7015625904065544847387 x + 0.7015625904065544847387 y + z)}^{6} - 2.787017736002609434885 {(0.8275051790378073440092 x + 0.8275051790378073440092 y - z)}^{6} - 18.14043459667601807177 {(0.8277341603719508826234 x - 0.3874078909018352984282 y + z)}^{6} - 18.14043459667601807177 {(0.8277341603719508826234 x + 0.3874078909018352984282 y - z)}^{6} + 18.17470588318876109328 {(0.8691668928268026710331 x - 9.839594044888732451897 \cdot 10^{-87} y + z)}^{6} + 18.17470588318876109328 {(0.8691668928268026710331 x - 7.09814488652441969896 \cdot 10^{-87} y - z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 7.119093909182977317174 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.0004783088843963595609995 \pm 6.1 \cdot 10^{-18} \end{array}

Degree 7

Degree 7, ⟨𝐽 ∐ 15⟩ (Harnack's M curve) 🟢

\begin{array}{rcl} P & = & - 1.74415402874978 \cdot 10^{-10} x^{7} + 0.193985061996207 x^{6} y - 0.0674698195570194 x^{6} z + 1.47457241517702 \cdot 10^{-8} x^{5} y^{2} - 3.34288587749802 \cdot 10^{-8} x^{5} y z + 1.01491178382314 \cdot 10^{-8} x^{5} z^{2} + 3.19653872230425 x^{4} y^{3} - 0.537105063338043 x^{4} y^{2} z - 1.12612133763269 x^{4} y z^{2} + 0.322178912498801 x^{4} z^{3} - 6.16682520044354 \cdot 10^{-8} x^{3} y^{4} - 2.28770197209962 \cdot 10^{-7} x^{3} y^{3} z + 1.98897825628962 \cdot 10^{-8} x^{3} y^{2} z^{2} + 1.10575615971594 \cdot 10^{-7} x^{3} y z^{3} - 3.04774765965544 \cdot 10^{-8} x^{3} z^{4} - 0.412696946812073 x^{2} y^{5} - 2.87922246275859 x^{2} y^{4} z - 4.69434441310397 x^{2} y^{3} z^{2} + 0.79744414055892 x^{2} y^{2} z^{3} + 1.81877730525204 x^{2} y z^{4} - 0.487300026163997 x^{2} z^{5} + 3.89755112946603 \cdot 10^{-9} x y^{6} + 4.20794560514053 \cdot 10^{-8} x y^{5} z + 1.50063760088956 \cdot 10^{-7} x y^{4} z^{2} + 1.73813500346809 \cdot 10^{-7} x y^{3} z^{3} - 3.22693680864886 \cdot 10^{-8} x y^{2} z^{4} - 7.91808693362777 \cdot 10^{-8} x y z^{5} + 2.17225891352177 \cdot 10^{-8} x z^{6} + 0.0132679769536452 y^{7} + 0.187064674904409 y^{6} z + 0.978383145670196 y^{5} z^{2} + 2.24737285428712 y^{4} z^{3} + 1.84740166166103 y^{3} z^{4} - 0.46881063416948 y^{2} z^{5} - 0.81032990850427 y z^{6} + 0.222620135268147 z^{7} \\ ≃ & 737.583632249725877922106 {(- x - 0.7430393317874477789192 y - 0.8783032079440896723802 z)}^{7} - 522.166745110871534332099 {(- x - 0.345201971887186757688792 y - 0.992400308938238450847721 z)}^{7} + 13.5338736804438272108341 {(- x - 0.032544020329245999475341 y + 0.702663067722345839900194 z)}^{7} + 1392.08351380700445739413 {(- x + 0.496890579078654594886053 y + 0.950997197283469082170708 z)}^{7} - 737.583529677060280996719 {(- x + 0.743039351238252417263789 y + 0.878303244045156127144227 z)}^{7} + 2712.37399705287891175064 {(- 0.810561602964861236527234 x + 0.347844590607047120409006 y + z)}^{7} - 2776.33933459518796115043 {(- 0.571963633287207716347078 x + y + 0.350841178780400211176522 z)}^{7} - 4023.11923601468166064837 {(- 0.558694059828629659752311 x + 0.344795883897282020108047 y + z)}^{7} - 7031.52375360478748348248 {(- 0.355160749944217022218366 x - y - 0.112982103724865975264171 z)}^{7} + 5877.09335304817717709326 {(- 0.278905248244801235366719 x + 0.347843056650947065641545 y + z)}^{7} + 12224.3119313938492766935 {(- 0.192898240460801982842019 x - y + 0.0670226346514183292531303 z)}^{7} - 17593.4136549058751564525 {(- 0.0754484589178065130048167 x - y + 0.19095631764005904426417 z)}^{7} + 6816.4262086221959207539 {(- 1.01449680877731271488031 \cdot 10^{-8} x - 0.349255322568968955579317 y - z)}^{7} + 5486.17016256516551167844 {(- 2.00869668293423156226093 \cdot 10^{-10} x + y - 0.272939103913952872448434 z)}^{7} + 25906.5374312034528745542 {(- 1.01336532310860083546528 \cdot 10^{-10} x - y + 0.240534781256189021570316 z)}^{7} - 15.7168985315477461884406 {(3.78709129858557151628194 \cdot 10^{-9} x - y + 0.657474836793920856895284 z)}^{7} - 17593.4136984766606689627 {(0.0754484577636922460966513 x - y + 0.190956316165219462095674 z)}^{7} + 12224.3120087950311132039 {(0.192898236847259869017996 x - y + 0.0670226309928145783016208 z)}^{7} + 5877.09313903143764815481 {(0.278905269979036241087271 x + 0.347843057152659693078279 y + z)}^{7} - 7031.52383557750175703877 {(0.355160742556181711798671 x - y - 0.112982110161238846998282 z)}^{7} - 4023.1189425435335787873 {(0.558694085919739795318808 x + 0.344795884870542444061626 y + z)}^{7} - 2776.33928247141214495846 {(0.571963646054381875496642 x + y + 0.350841169052981428518166 z)}^{7} + 2712.37370999858769567625 {(0.810561635502876765418596 x + 0.347844592065160428261279 y + z)}^{7} - 661.484304215640092329288 {(0.878865467022532596709758 x - y - 0.682048508368015631468337 z)}^{7} + 661.484285133070808973137 {(0.878865488055371175762438 x + y + 0.682048494786057242573682 z)}^{7} + 2379.42098829348197721524 {(0.973785922083014070403365 x - 0.379977871411113311314881 y - z)}^{7} - 2379.42068576712383386505 {(0.973785960204145883455996 x + 0.379977873746494947744039 y + z)}^{7} - 522.166671535363017454661 {(x - 0.345201983524998599589661 y - 0.992400347566657150202895 z)}^{7} + 13.5338749076322683399112 {(x - 0.0325440245968627515113304 y + 0.702663039968101975813966 z)}^{7} + 1392.08370936341937722393 {(x + 0.496890564417769086891484 y + 0.950997159546439708772479 z)}^{7} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 9.3661690299577384587753 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.00000238287955329906566192218 \pm 2.8 \cdot 10^{-17} \end{array}

Degree 8

Degree 8, ⟨18 ∐ 1⟨3⟩⟩ (Harnack's M curve) 🟢

\begin{array}{rcl} P & = & 3.42639808625191 \cdot 10^{-8} x^{8} - 5.20218194718691 \cdot 10^{-9} x^{7} y - 3.34927900961968 \cdot 10^{-9} x^{7} z + 2.14927084494929 x^{6} y^{2} + 2.76750793316963 x^{6} y z + 0.890891344032204 x^{6} z^{2} - 8.7090761580403 \cdot 10^{-9} x^{5} y^{3} + 1.99506657847714 \cdot 10^{-8} x^{5} y^{2} z + 5.118016639116 \cdot 10^{-8} x^{5} y z^{2} + 2.23976367917429 \cdot 10^{-8} x^{5} z^{3} + 0.716423769876259 x^{4} y^{4} + 4.6125132170289 x^{4} y^{3} z + 1.14924961321223 x^{4} y^{2} z^{2} - 3.52509693886025 x^{4} y z^{3} - 1.63800689164346 x^{4} z^{4} - 1.75990922440709 \cdot 10^{-9} x^{3} y^{5} - 8.52627334429744 \cdot 10^{-9} x^{3} y^{4} z + 4.80459271614805 \cdot 10^{-8} x^{3} y^{3} z^{2} + 3.33511890999359 \cdot 10^{-8} x^{3} y^{2} z^{3} - 3.73752474256241 \cdot 10^{-8} x^{3} y z^{4} - 2.39056725244219 \cdot 10^{-8} x^{3} z^{5} - 1.194039148444 x^{2} y^{6} + 0.922502623527613 x^{2} y^{5} z + 3.68829033047973 x^{2} y^{4} z^{2} - 2.35006459612284 x^{2} y^{3} z^{3} - 3.27601379108108 x^{2} y^{2} z^{4} + 1.14917690065212 x^{2} y z^{5} + 1.02387845537451 x^{2} z^{6} + 1.74698495308752 \cdot 10^{-9} x y^{7} - 7.23401247549844 \cdot 10^{-9} x y^{6} z - 3.17088271888009 \cdot 10^{-9} x y^{5} z^{2} + 2.62547268472772 \cdot 10^{-8} x y^{4} z^{3} - 5.24450473680407 \cdot 10^{-9} x y^{3} z^{4} - 2.39160381153039 \cdot 10^{-8} x y^{2} z^{5} + 4.32834637838548 \cdot 10^{-9} x y z^{6} + 7.1086594668879 \cdot 10^{-9} x z^{7} + 0.23880789236505 y^{8} - 0.922502642618129 y^{7} z + 0.721621944432765 y^{6} z^{2} + 1.17503233171612 y^{5} z^{3} - 1.63800687629417 y^{4} z^{4} - 0.383058989845222 y^{3} z^{5} + 1.02387845225685 y^{2} z^{6} + 5.12029721743252 \cdot 10^{-9} y z^{7} - 0.215868035709408 z^{8} \\ ≃ & - 41092.753101152946135664631 {(- x - 0.78377816208381896545425688 y + 0.73965084069786299859205662 z)}^{8} + 101688.20790401539286398679 {(- x - 0.51112945346182100011645246 y + 0.79558758477653398332032514 z)}^{8} + 272236.89420214674728393621 {(- x - 0.26866765768780414913863019 y + 0.41818887756115697704011554 z)}^{8} + 272236.89076910931133645488 {(- x + 0.2686676581158176183230246 y - 0.41818888198682753313923478 z)}^{8} - 153180.37509889329580389426 {(- x + 0.40222910364351236450615708 y - 0.62747585878708633683521774 z)}^{8} + 101688.20546442076818924378 {(- x + 0.51112945499913986400789293 y - 0.79558759092887005607543752 z)}^{8} + 573.82086707385807197495238 {(- 0.91247685013667526444729847 x + 0.52681875750222005432955982 y - z)}^{8} + 158964.06012228293637267103 {(- 0.57735026918661909382640946 x + y + 2.6918972126833281548672355 \cdot 10^{-10} z)}^{8} - 243453.63867355946852148606 {(- 0.41304212435818333218493726 x + y + 0.22113039066831765695352774 z)}^{8} + 273020.24625132097104870222 {(- 0.26723098646731015913584743 x + y + 0.41803870815880216665364816 z)}^{8} - 257616.79657405273269001908 {(- 0.14211761058350007712286153 x + y + 0.58799771807245375231205929 z)}^{8} - 257616.79657273177327057023 {(- 0.14211760837342227664952697 x - y - 0.58799771860811310049052458 z)}^{8} + 254650.19728016590324178957 {(- 0.05113178954334642829357377 x + y + 0.70933897257945121066257308 z)}^{8} + 254650.19727969611471613688 {(- 0.051131786876160487588622976 x - y - 0.70933897277220140651781269 z)}^{8} + 62474.945637616317852239657 {(- 1.4634015991045443396840104 \cdot 10^{-9} x + y + 0.77826427910536737490115777 z)}^{8} - 327499.46033021113115662868 {(1.4204809541986528049259963 \cdot 10^{-9} x - y - 0.75547339842913188206890986 z)}^{8} + 871.57773423290117543935187 {(1.78511270603536564397396 \cdot 10^{-9} x - y - 0.9490930135373968937752746 z)}^{8} - 20.09668559988582227673035 {(1.8849436086770641560948949 \cdot 10^{-9} x + 0.75651455491123575518210028 y - z)}^{8} + 273020.24624868858806716354 {(0.26723098489760975574256869 x + y + 0.41803870916582490776881038 z)}^{8} - 243453.63866993137877067856 {(0.41304212353058060192301557 x + y + 0.22113039222444255390801505 z)}^{8} + 158964.06011897158149883914 {(0.57735026919263243420661713 x + y + 2.4437652668362515602945954 \cdot 10^{-9} z)}^{8} - 20.096686045729347223753604 {(0.65516081801184222361669681 x + 0.37825727437047294989026441 y + z)}^{8} - 20.096685648997123923518326 {(0.65516082339331728810350926 x - 0.37825727530683318645972632 y - z)}^{8} - 71304.952265795933478619423 {(0.76891747478018995264070886 x - y + 0.25781637885680186933046808 z)}^{8} - 71304.95226381774881049035 {(0.76891747575842545799966033 x + y - 0.25781637596158723009903623 z)}^{8} + 573.82085129689887742654548 {(0.91247685704154667331997448 x + 0.52681875930868684148433881 y - z)}^{8} + 22606.251390899473160342848 {(x - 0.9986604666328049374294719 y + 0.57077150597894813267481413 z)}^{8} - 41092.752184156171237602758 {(x - 0.78377816427460879780713969 y + 0.73965084652752889889957734 z)}^{8} + 63398.411897559632777194184 {(x - 0.64759982251797063795020648 y + 0.84398946578576729645964656 z)}^{8} + 19767.46316725915625831158 {(x - 0.57735027138875186166523813 y + 0.89866218219140255852657055 z)}^{8} - 103622.87560667255350504408 {(x - 0.57735027132444016546814445 y + 0.87234553997136114343871012 z)}^{8} - 426329.31707296075435436071 {(x - 0.13267026916534226112727953 y + 0.20617325123822298269600597 z)}^{8} + 502404.93075161207419456861 {(x - 2.2550045585050086749380026 \cdot 10^{-12} y + 1.8832376649071639317685996 \cdot 10^{-9} z)}^{8} - 426329.31972351766144087686 {(x + 0.13267026905772824209846286 y - 0.20617324731152118406435103 z)}^{8} - 153180.37799729442944874343 {(x + 0.4022291026876555552377843 y - 0.62747585353651365246914212 z)}^{8} - 103622.87833259402512686151 {(x + 0.57735026942144554479490381 y - 0.87234553333637658745334096 z)}^{8} + 19767.463702940459802742603 {(x + 0.57735026942852971433156211 y - 0.89866217538080545153828046 z)}^{8} + 63398.413511321174950951057 {(x + 0.64759982045293647284234237 y - 0.84398945933389796666359291 z)}^{8} + 22606.251780504791963144767 {(x + 0.99866046447687993375220631 y - 0.57077150098285932219238387 z)}^{8} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 7.1837734110860747196532771 \cdot 10^{-17} \\ dist (P, Δ) & = & 3.4263980859463065386598227 \cdot 10^{-8} \pm 1.9 \cdot 10^{-17} \end{array}

Degree 8, ⟨9 ∐ 1⟨1⟨1⟩⟩⟩ (depth 3 nest) 🟢

\begin{array}{rcl} P & = & 0.00184852034526133 x^{8} + 1.37962114569464 \cdot 10^{-10} x^{7} y + 2.70336709326394 \cdot 10^{-11} x^{7} z + 1.80339843306263 x^{6} y^{2} + 1.33598915259897 x^{6} y z + 0.139098862673101 x^{6} z^{2} + 2.33771699033518 \cdot 10^{-10} x^{5} y^{3} - 2.9149896713279 \cdot 10^{-9} x^{5} y^{2} z - 3.09849552780582 \cdot 10^{-9} x^{5} y z^{2} - 3.79596195179658 \cdot 10^{-10} x^{5} z^{3} + 0.609759239170446 x^{4} y^{4} + 2.22664858868118 x^{4} y^{3} z - 8.60811680882531 x^{4} y^{2} z^{2} - 7.27611724198381 x^{4} y z^{3} - 0.708656388993302 x^{4} z^{4} + 4.85439322485681 \cdot 10^{-11} x^{3} y^{5} + 1.08627261138854 \cdot 10^{-9} x^{3} y^{4} z - 2.45539034487757 \cdot 10^{-9} x^{3} y^{3} z^{2} + 2.98701581111141 \cdot 10^{-9} x^{3} y^{2} z^{3} + 4.49579850208009 \cdot 10^{-9} x^{3} y z^{4} + 4.80588578830014 \cdot 10^{-10} x^{3} z^{5} - 0.990386114083156 x^{2} y^{6} + 0.445329716098042 x^{2} y^{5} z + 6.4342388546789 x^{2} y^{4} z^{2} - 4.85074482503623 x^{2} y^{3} z^{3} - 1.41731277900454 x^{2} y^{2} z^{4} + 2.82624465449354 x^{2} y z^{5} + 0.299067689378958 x^{2} z^{6} - 4.72656519272849 \cdot 10^{-11} x y^{7} + 2.74110927805355 \cdot 10^{-11} x y^{6} z + 6.68800018719502 \cdot 10^{-10} x y^{5} z^{2} - 1.63868960717203 \cdot 10^{-9} x y^{4} z^{3} + 5.18990151969059 \cdot 10^{-10} x y^{3} z^{4} + 4.83605887668308 \cdot 10^{-10} x y^{2} z^{5} - 5.87765608693515 \cdot 10^{-10} x y z^{6} - 6.37339336336658 \cdot 10^{-11} x z^{7} + 0.201404559463765 y^{8} - 0.445329717212565 y^{7} z - 0.863724849128429 y^{6} z^{2} + 2.42537241348016 y^{5} z^{3} - 0.708656387256309 y^{4} z^{4} - 0.942081551942447 y^{3} z^{5} + 0.299067688971784 y^{2} z^{6} + 4.41515556763352 \cdot 10^{-11} y z^{7} - 0.00184852034526133 z^{8} \\ ≃ & 0.748889502883901619695 {(- x - 0.956287709064390712351 y + 0.617228908952546817997 z)}^{8} + 0.221873659422060425097 {(- x - 0.577350269310259400498 y + 0.91071735282544279893 z)}^{8} - 2.7506880105010065849 {(- x - 0.577350269272726597886 y + 0.626527055916616037026 z)}^{8} + 7.25179692389312110635 {(- x - 0.292042216132865199932 y + 0.464721507749055561345 z)}^{8} + 2.59849477318757139295 {(- x - 0.0533515135520429345353 y + 0.446722006393310612235 z)}^{8} + 19.0449919807034076518 {(- x - 2.66900892274135847218 \cdot 10^{-13} y + 1.03983497640828949957 \cdot 10^{-10} z)}^{8} - 17.3752970684814084603 {(- x + 0.100957873701140638522 y - 0.276608227818251225118 z)}^{8} - 2.75068801337503852489 {(- x + 0.577350269196787883441 y - 0.626527055626821388002 z)}^{8} + 0.221873659758788922833 {(- x + 0.57735026920019801374 y - 0.910717352444706035382 z)}^{8} - 0.497984836773092874302 {(- 0.757867566652633135267 x + 0.107545126493868846069 y - z)}^{8} - 0.49798483740099773496 {(- 0.757867566325160133248 x - 0.107545126477323054811 y + z)}^{8} - 3.40033670619497299046 {(- 0.720292597774072445198 x - y + 0.339169181824907872556 z)}^{8} + 0.64012552483783262938 {(- 0.650746405748616915991 x - y + 0.532223975829907027727 z)}^{8} + 6.02595449388700900722 {(- 0.577350269189981632357 x + y + 1.32069261410581927106 \cdot 10^{-10} z)}^{8} - 8.64984743642318684341 {(- 0.450153805526568002565 x - y - 0.301807875538753859815 z)}^{8} - 0.497984837203297852693 {(- 0.285796971812549344997 x - 0.710105128397777927447 y - z)}^{8} - 0.497984837440085011445 {(- 0.285796971587974679843 x + 0.710105128355419320467 y + z)}^{8} + 7.98085485770896170382 {(- 0.244142951002394298809 x - y - 0.459189871488473603833 z)}^{8} + 7.980854857700640932 {(- 0.244142950907463580422 x + y + 0.459189871539307123197 z)}^{8} + 0.701230331226329321221 {(- 8.17453373980676734132 \cdot 10^{-11} x - y - 0.788704363202146166973 z)}^{8} - 8.69353247945619696294 {(- 5.61533331648754730174 \cdot 10^{-11} x - y - 0.542588346585836819534 z)}^{8} + 11.8899432155100874921 {(1.03931029906658516489 \cdot 10^{-10} x + 0.19658133670267576122 y + z)}^{8} - 32.9186456805313490273 {(1.03983497640848123959 \cdot 10^{-10} x - 7.2034361056322376643 \cdot 10^{-11} y + z)}^{8} + 11.8899432151735255242 {(0.170244431453395696874 x + 0.0982906684597826429513 y - z)}^{8} + 11.8899432118057968068 {(0.170244431667442701935 x - 0.0982906684631717691952 y + z)}^{8} - 8.64984743640655886769 {(0.450153805464443896077 x - y - 0.301807875632443516449 z)}^{8} - 0.49798483714054668915 {(0.472070594661317510722 x + 0.602560002140964111157 y - z)}^{8} - 0.497984836749428987238 {(0.472070594915951732951 x - 0.602560002199868509404 y + z)}^{8} + 1.04802258683287440813 {(0.508340589204572950905 x - y - 0.500416069951499075832 z)}^{8} + 1.04802258683514948167 {(0.508340589307971235767 x + y + 0.500416069845645221446 z)}^{8} + 6.0259544939018661477 {(0.577350269189269896659 x + y + 1.19994607021369070388 \cdot 10^{-11} z)}^{8} + 0.640125524836053748238 {(0.650746405860061788345 x - y + 0.532223975694758131448 z)}^{8} - 3.40033670618451373226 {(0.720292597845419190406 x - y + 0.339169181675241193603 z)}^{8} + 0.748889503655998966933 {(x - 0.956287708940616566015 y + 0.617228908665035243534 z)}^{8} + 7.25179692950906512204 {(x - 0.29204221610406094407 y + 0.464721507496102302726 z)}^{8} + 2.59849477511943610489 {(x - 0.0533515135465510745617 y + 0.446722006143828886553 z)}^{8} - 17.3752970604777526022 {(x + 0.100957873707487521331 y - 0.276608228042145121215 z)}^{8} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 5.56769886681425493263 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.00184852034526132908152 \pm 1.6 \cdot 10^{-17} \end{array}

Degree 8, ⟨19⟩ (M-3 curve) 🟢

This curve has 43 quasi-critical points, but the rank of the corresponding family of 8th power of linear forms is only 41, hence the polynomial is written using only 41 independent 8th power of linear forms. The missing points appear clearly on the drawing.

\begin{array}{rcl} P & = & 0.0495589143528351 x^{8} - 0.00188018680131159 x^{7} y - 0.000795119640932293 x^{7} z - 0.98222967758031 x^{6} y^{2} - 0.512485805811381 x^{6} y z - 0.387887731197622 x^{6} z^{2} + 0.0196278020389989 x^{5} y^{3} + 0.0177437997933578 x^{5} y^{2} z + 0.0131094554566617 x^{5} y z^{2} + 0.00388115481719391 x^{5} z^{3} + 4.81286215768501 x^{4} y^{4} + 4.37041111569147 x^{4} y^{3} z + 5.46367682835569 x^{4} y^{2} z^{2} + 2.26805542914645 x^{4} y z^{3} + 0.96347855083981 x^{4} z^{4} - 0.0115396380424817 x^{3} y^{5} - 0.00891660576928047 x^{3} y^{4} z - 0.0238949884583324 x^{3} y^{3} z^{2} - 0.0108324040428585 x^{3} y^{2} z^{3} - 0.0125477647548797 x^{3} y z^{4} - 0.00487482221080735 x^{3} z^{5} + 0.245163497293747 x^{2} y^{6} + 3.32420020860353 x^{2} y^{5} z - 1.75134056243491 x^{2} y^{4} z^{2} + 2.90577734180137 x^{2} y^{3} z^{3} - 1.78664997060123 x^{2} y^{2} z^{4} - 2.14515883780769 x^{2} y z^{5} - 0.882233333049585 x^{2} z^{6} - 0.000352126771039669 x y^{7} - 0.00481778746941958 x y^{6} z - 0.000622356910905359 x y^{5} z^{2} - 0.00303919507158106 x y^{4} z^{3} + 0.00499500693115176 x y^{3} z^{4} - 0.000908018084730323 x y^{2} z^{5} + 0.0030737225764952 x y z^{6} + 0.00177576523890042 x z^{7} - 0.000593764825451007 y^{8} + 0.00802584243368985 y^{7} z + 0.121879421524613 y^{6} z^{2} - 1.1278120374435 y^{5} z^{3} + 2.6700988140946 y^{4} z^{4} - 1.70424194371066 y^{3} z^{5} - 0.740965215107721 y^{2} z^{6} + 0.511633629359521 y z^{7} + 0.262270902028137 z^{8} \\ ≃ & 1.390023128461474579 {(- x - 0.514864672901870670517 y + 0.694792149280820947404 z)}^{8} - 5.76856376174915765797 {(- x - 0.367651952864542142082 y + 0.449029247862737071282 z)}^{8} + 11.0189889580600322248 {(- x - 0.348221060805872479425 y + 0.162547771489192290354 z)}^{8} + 5.28914333657159287355 {(- x - 0.0623240371443920724944 y + 0.863300220608303397447 z)}^{8} - 13.6179238115978697977 {(- x - 0.0133017772624531407553 y + 0.580884258432679159669 z)}^{8} - 12.9799982438379493272 {(- x + 0.0147305009770848275137 y - 0.57992073239483549944 z)}^{8} - 15.0428544729854527006 {(- x + 0.286420392192215051987 y + 0.106533249699119379954 z)}^{8} - 8.21674557054554787186 {(- 0.832142048755301378634 x - 0.0523334213113831989111 y - z)}^{8} - 8.11552529248380784848 {(- 0.830832358180963667038 x + 0.053493992183472768816 y + z)}^{8} + 3.43594288028844556097 {(- 0.693008211955235835485 x - y + 0.596367698528421295515 z)}^{8} + 3.5179481547302908655 {(- 0.691321266081178478422 x + y - 0.595271246521781144432 z)}^{8} + 9.12166752574641419074 {(- 0.627558131122870499815 x - 0.333099558503366360046 y - z)}^{8} + 9.03462017052236159039 {(- 0.626063742643032917838 x + 0.33384616536963884814 y + z)}^{8} - 6.80361606701100434313 {(- 0.42602969155936136302 x + 0.616319883065497222444 y + z)}^{8} - 18.7234703598625914949 {(- 0.381994900341736507673 x - y + 0.317287446742690586326 z)}^{8} - 18.9774784807734199622 {(- 0.380583277447180849589 x + y - 0.316835905923849674503 z)}^{8} + 3.69781507743267734732 {(- 0.215922762999589999033 x - 0.895930219341788753721 y - z)}^{8} - 5.50865556312886726439 {(- 0.0010471814359080330952 x - y - 0.904757201368224393515 z)}^{8} - 100.113503296839096484 {(- 0.000708485063682819988933 x - y + 0.0238613644964277896563 z)}^{8} + 33.9217223339916592146 {(0.000769903958928277064165 x + y + 0.144533490296711020538 z)}^{8} - 8.8617659631969015615 {(0.000852723761545914622879 x + y + 0.371604143066634607299 z)}^{8} + 5.01134119110655147787 {(0.000949962549676449763213 x + y + 0.638207955168146082825 z)}^{8} + 52.1382396941048274585 {(0.160693766253681369832 x - y + 0.129247334735093325364 z)}^{8} + 51.8119431496453608733 {(0.162071164685063605015 x + y - 0.129394992845831119458 z)}^{8} + 3.67917075679684490081 {(0.213874391306113097406 x - 0.896098005017882744826 y - z)}^{8} - 6.85606322258123997565 {(0.427775829028201283447 x + 0.615899406602238563444 y + z)}^{8} - 0.168158090550749774952 {(0.715322446548715959756 x - 0.372214115513973854825 y + z)}^{8} + 2.19386791269369689742 {(0.828811689671995075658 x + 0.195574543656054599001 y - z)}^{8} + 2.37200760180158879011 {(0.829761350398868711896 x - 0.19688275752418034993 y + z)}^{8} + 0.0713041294826755666898 {(0.936821208170425277126 x - 0.570529403619022106307 y + z)}^{8} - 0.980972440487592667718 {(x - 0.834524192744189608891 y + 0.869136844752565805157 z)}^{8} + 1.32947040636650824212 {(x - 0.516419196823238992133 y + 0.694224511242661041957 z)}^{8} - 5.07900564980871701715 {(x - 0.369160421308773292273 y + 0.448390027651836735607 z)}^{8} + 10.5872881566921501478 {(x - 0.349788852778525302644 y + 0.161880319012975976822 z)}^{8} + 2.53494813346077842487 {(x - 0.220157481488407112402 y + 0.590156923216909136977 z)}^{8} + 15.583818312096740699 {(x - 0.159136773462445597936 y - 0.351210389544920435209 z)}^{8} + 5.25385219757797448039 {(x - 0.0637244489218174766948 y + 0.862105839372452686507 z)}^{8} + 15.2447221886636640524 {(x + 0.157625892024054788088 y + 0.350311493330204895941 z)}^{8} + 3.25367506160861802164 {(x + 0.218748564203778468505 y - 0.590954896662565830591 z)}^{8} - 15.1015917993935178387 {(x + 0.284846874561262385414 y + 0.10575208127168468563 z)}^{8} - 0.953137039630366940463 {(x + 0.832621710691360088755 y - 0.869378162435763103742 z)}^{8} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.37922513912681184877 \cdot 10^{-16} \\ dist (P, Δ) & = & 0.00069817037013123745928 \pm 1.2 \cdot 10^{-17} \end{array}

Degree 9

Degree 9 ⟨J U 28⟩ (Harnack's M curve) 🔴

\begin{array}{rcl} P & = & - 6.47690131545843 \cdot 10^{-14} x^{9} - 0.000894699521756603 x^{8} y - 0.000896140509912845 x^{8} z - 3.09220644734732 \cdot 10^{-10} x^{7} y^{2} - 5.66814054893491 \cdot 10^{-10} x^{7} y z - 2.57068972868567 \cdot 10^{-10} x^{7} z^{2} - 1.51675134076296 x^{6} y^{3} - 4.20115750163369 x^{6} y^{2} z - 3.84483992258021 x^{6} y z^{2} - 1.16042305503365 x^{6} z^{3} - 2.98903753395465 \cdot 10^{-11} x^{5} y^{4} - 6.37583262041529 \cdot 10^{-10} x^{5} y^{3} z + 6.4130497948779 \cdot 10^{-10} x^{5} y^{2} z^{2} + 2.52686110460851 \cdot 10^{-9} x^{5} y z^{3} + 1.27602721429944 \cdot 10^{-9} x^{5} z^{4} + 1.51257607600666 x^{4} y^{5} - 1.40456782361952 x^{4} y^{4} z - 6.40806653727669 x^{4} y^{3} z^{2} + 0.586067739569115 x^{4} y^{2} z^{3} + 7.18995079268125 x^{4} y z^{4} + 3.11292954998475 x^{4} z^{5} + 1.22179986633867 \cdot 10^{-10} x^{3} y^{6} - 2.1265510462967 \cdot 10^{-10} x^{3} y^{5} z - 1.11937042997033 \cdot 10^{-9} x^{3} y^{4} z^{2} + 1.75169667221458 \cdot 10^{-9} x^{3} y^{3} z^{3} + 2.10856613962517 \cdot 10^{-9} x^{3} y^{2} z^{4} - 1.8126185907653 \cdot 10^{-9} x^{3} y z^{5} - 1.38122584014615 \cdot 10^{-9} x^{3} z^{6} - 0.504788491576338 x^{2} y^{7} + 2.32840040407918 x^{2} y^{6} z - 1.28161330874381 x^{2} y^{5} z^{2} - 6.19282709927595 x^{2} y^{4} z^{3} + 4.79330053000986 x^{2} y^{3} z^{4} + 6.22585909895079 x^{2} y^{2} z^{5} - 2.73630046153379 x^{2} y z^{6} - 2.0907856951756 x^{2} z^{7} - 2.61224068948139 \cdot 10^{-11} x y^{8} + 1.62398508872973 \cdot 10^{-10} x y^{7} z - 1.87125656101384 \cdot 10^{-10} x y^{6} z^{2} - 5.99291736462113 \cdot 10^{-10} x y^{5} z^{3} + 1.26840898132158 \cdot 10^{-9} x y^{4} z^{4} + 2.08237515797689 \cdot 10^{-10} x y^{3} z^{5} - 1.33685646372163 \cdot 10^{-9} x y^{2} z^{6} + 1.30498729084796 \cdot 10^{-10} x y z^{7} + 1.79970366202169 \cdot 10^{-10} x z^{8} + 0.0563858433370229 y^{9} - 0.467293133613127 y^{8} z + 1.28161330741715 y^{7} z^{2} - 0.708496733216063 y^{6} z^{3} - 2.39665026294871 y^{5} z^{4} + 3.11292955057919 y^{4} z^{5} + 0.912100152380785 y^{3} z^{6} - 2.09078569526425 y^{2} z^{7} + 1.40425959287793 \cdot 10^{-10} y z^{8} + 0.00705483675959769 z^{9} \\ ≃ & 2380896.82519547 {(- x - 0.729258346206677 y + 0.727781732441915 z)}^{9} - 6947676.05364025 {(- x - 0.619721382928701 y + 0.618725694861009 z)}^{9} + 13220124.7356317 {(- x - 0.577350269174812 y + 0.584631971179082 z)}^{9} - 9609406.44174142 {(- x - 0.536478908104933 y + 0.596825504671969 z)}^{9} + 7241456.37737902 {(- x - 0.44310322926996 y + 0.643168848479633 z)}^{9} + 6821.3739251457 {(- x - 0.268061626727725 y + 0.293770854043093 z)}^{9} + 3112667.81040602 {(- x - 0.0695426730080776 y + 0.829962798402872 z)}^{9} + 9609406.44632266 {(- x + 0.53647890810518 y - 0.596825504526279 z)}^{9} - 13220124.7417725 {(- x + 0.577350269174157 y - 0.584631971033646 z)}^{9} + 2603883.3093554 {(- x + 0.577350269182715 y - 0.576422665048878 z)}^{9} - 2380896.82657527 {(- x + 0.729258346188698 y - 0.72778173227943 z)}^{9} + 380834.337196557 {(- x + 0.96793336044347 y - 0.96637815793565 z)}^{9} - 412.346899407536 {(- x + 0.999790222623675 y - 0.401245079295575 z)}^{9} + 968264.399786432 {(- 0.702257600040002 x - 0.786872848026557 y - z)}^{9} + 589843.548169065 {(- 0.673952454347285 x - y + 0.998446601453092 z)}^{9} - 5671750.2421848 {(- 0.250560639046082 x + y + 0.715839869966238 z)}^{9} + 15415801.7564314 {(- 0.106899414664023 x + y + 0.591377748152394 z)}^{9} + 29862547.5844375 {(- 0.0312057953677463 x - y - 0.526178255824751 z)}^{9} - 29862547.5845394 {(- 0.0312057953401219 x + y + 0.526178255826924 z)}^{9} - 1175698.57508409 {(- 4.40385294436983 \cdot 10^{-11} x + y - 0.998366122433211 z)}^{9} - 48245839.5044969 {(- 1.26525246074838 \cdot 10^{-11} x - y - 0.506306138757093 z)}^{9} + 730564.473970961 {(- 1.04368097832507 \cdot 10^{-11} x - y - 0.463005198034606 z)}^{9} - 9502674.04500366 {(1.22366877958784 \cdot 10^{-11} x + y + 0.499196671225386 z)}^{9} + 15415801.7562901 {(0.106899414699384 x + y + 0.591377748145064 z)}^{9} - 5671750.24205332 {(0.250560639096163 x + y + 0.715839869949062 z)}^{9} - 6824.8260086732 {(0.267836764427584 x - y - 0.29375433989396 z)}^{9} + 6824.82600799203 {(0.267836764437262 x + y + 0.293754339867268 z)}^{9} + 982379.469375189 {(0.329792330460547 x + y - 0.998393240386254 z)}^{9} - 982379.469458807 {(0.329792330545524 x - y + 0.998393240338582 z)}^{9} - 1215553.36513285 {(0.488205894752592 x - y - 0.921365232844725 z)}^{9} + 1215553.3650689 {(0.488205894826941 x + y + 0.921365232811234 z)}^{9} + 589843.548271855 {(0.67395245442239 x - y + 0.998446601355671 z)}^{9} + 968264.400301596 {(0.702257599914497 x - 0.786872847989835 y - z)}^{9} - 1158522.63341402 {(0.867442699517211 x - 0.500818274816218 y - z)}^{9} + 1158522.63260817 {(0.867442699672619 x + 0.500818274840943 y + z)}^{9} - 6947676.05706318 {(x - 0.619721382923902 y + 0.618725694711667 z)}^{9} - 200186.245600638 {(x - 0.57735026918748 y + 0.534632351471496 z)}^{9} + 7241456.38116333 {(x - 0.443103229271273 y + 0.643168848331481 z)}^{9} - 5243069.58284678 {(x - 0.285490253090748 y + 0.722118164843785 z)}^{9} + 6821.37392682709 {(x - 0.268061626751556 y + 0.293770853921731 z)}^{9} + 1292140.98634919 {(x - 0.207960989785132 y - 0.968447332350232 z)}^{9} + 3112667.81254989 {(x - 0.0695426730240845 y + 0.829962798239425 z)}^{9} - 1292140.98737471 {(x + 0.20796098974824 y + 0.968447332170496 z)}^{9} + 5243069.57971777 {(x + 0.285490253085343 y - 0.722118164997361 z)}^{9} + 2603883.30817163 {(x + 0.577350269182815 y - 0.57642266519353 z)}^{9} + 200186.245510888 {(x + 0.57735026918518 y - 0.534632351612168 z)}^{9} + 380834.336903811 {(x + 0.96793336049731 y - 0.966378158133954 z)}^{9} - 412.346899225847 {(x + 0.999790222637554 y - 0.401245079429082 z)}^{9} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 0.695106665257518 \\ dist (P, Δ) & = & 2.4519823671465 \cdot 10^{-10} \pm 3.3 \cdot 10^{-14} \end{array}

Exceptional polynomials

Table of curves (see the lexicon below for explanation of the remarks)

Lexicon

Bibliography

Degree 2 to 5

Degree 2, ⟨1⟩ (M curve) 🟢Copy python definition

Degree 3, ⟨J ∐ 1⟩ (M curve) 🟢Copy python definition

Degree 4, ⟨4⟩ (M curve) 🟢Copy python definition

Degree 4, ⟨3⟩ (M-1 curve) 🟢Copy python definition

Degree 5, ⟨𝐽 ∐ 6⟩ (Harnack's M-curve) 🟢Copy python definition

Degree 5, ⟨𝐽 ∐ 5⟩ (M-1 curve) 🟢Copy python definition

Degree 6

M curve

Degree 6, ⟨9 ∐ 1⟨1⟩⟩ (Harnack's M-curve) 🟢Copy python definition

Degree 6, ⟨5 ∐ 1⟨5⟩⟩ (Gudkov's M curve) 🟢Copy python definition

Degree 6, ⟨1 ∐ 1⟨9⟩⟩ (Hilbert's M curve) 🟢Copy python definition

M-1 curve

Degree 6, ⟨10⟩ V1 (LM-1 curve) 🟢Copy python definition

Degree 6, ⟨10⟩ V2 (LM-1 curve) 🟢Copy python definition

Degree 6, ⟨8 ∐ 1⟨1⟩⟩ ( M-1 curve) 🟢Copy python definition

Degree 6, ⟨5 ∐ 1⟨4⟩⟩ (M-1 curve) 🟢Copy python definition

Degree 6, ⟨4 ∐ 1⟨5⟩⟩ (M-1 curve) 🔴Copy python definition

Degree 6, ⟨1 ∐ 1⟨8⟩⟩ (M-1 curve) 🟢Copy python definition

Degree 6, ⟨1⟨9⟩⟩ (M-1 curve) 🟢Copy python definition

M-2 curve

Degree 6, ⟨9⟩ V1 (M-2 curve) 🟢Copy python definition

Degree 6, ⟨9⟩ V2 (M-2 curve) 🟢Copy python definition

Degree 6, ⟨7 ∐ 1⟨1⟩⟩ (M-2 curve) 🟢Copy python definition

Degree 6, ⟨6 ∐ 1⟨2⟩⟩ (EM-2 curve) 🟢Copy python definition

Degree 6, ⟨5 ∐ 1⟨3⟩⟩ (M-2 curve) 🟢Copy python definition

Degree 6, ⟨4 ∐ 1⟨4⟩⟩ (M-2 curve) 🟢Copy python definition

Degree 6, ⟨3 ∐ 1⟨5⟩⟩ (M-2 curve) 🔴Copy python definition

Degree 6, ⟨2 ∐ 1⟨6⟩⟩ (EM-2 curve) 🟢Copy python definition

Degree 6, ⟨1 ∐ 1⟨7⟩⟩ (M-2 curve) 🟢Copy python definition

Degree 6, ⟨1⟨8⟩⟩ ND (M-2 curve) 🟢Copy python definition

Degree 6, ⟨1⟨8⟩⟩ D (M-2 curve) 🟢Copy python definition