Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + 1)y = x^6 - 6x^5 + 36x^3 - 51x^2 + 12x - 1$ | (homogenize, simplify) |
| $y^2 + (x^3 + z^3)y = x^6 - 6x^5z + 36x^3z^3 - 51x^2z^4 + 12xz^5 - z^6$ | (dehomogenize, simplify) |
| $y^2 = 5x^6 - 24x^5 + 146x^3 - 204x^2 + 48x - 3$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 12, -51, 36, 0, -6, 1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 12, -51, 36, 0, -6, 1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([-3, 48, -204, 146, 0, -24, 5]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(700569\) | \(=\) | \( 3^{6} \cdot 31^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(700569\) | \(=\) | \( 3^{6} \cdot 31^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(14236\) | \(=\) | \( 2^{2} \cdot 3559 \) |
| \( I_4 \) | \(=\) | \(56241\) | \(=\) | \( 3^{3} \cdot 2083 \) |
| \( I_6 \) | \(=\) | \(268101783\) | \(=\) | \( 3^{2} \cdot 401 \cdot 74287 \) |
| \( I_{10} \) | \(=\) | \(369024\) | \(=\) | \( 2^{7} \cdot 3 \cdot 31^{2} \) |
| \( J_2 \) | \(=\) | \(10677\) | \(=\) | \( 3 \cdot 3559 \) |
| \( J_4 \) | \(=\) | \(4728840\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 157 \cdot 251 \) |
| \( J_6 \) | \(=\) | \(2779512736\) | \(=\) | \( 2^{5} \cdot 7 \cdot 11 \cdot 13 \cdot 19 \cdot 4567 \) |
| \( J_8 \) | \(=\) | \(1828732434168\) | \(=\) | \( 2^{3} \cdot 3 \cdot 14783 \cdot 5154379 \) |
| \( J_{10} \) | \(=\) | \(700569\) | \(=\) | \( 3^{6} \cdot 31^{2} \) |
| \( g_1 \) | \(=\) | \(571005037946241799/2883\) | ||
| \( g_2 \) | \(=\) | \(71058711666950120/8649\) | ||
| \( g_3 \) | \(=\) | \(35206645259802016/77841\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
This curve has no rational points.
magma: []; // minimal model
magma: []; // simplified model
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable everywhere.
magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
Mordell-Weil group of the Jacobian
Group structure: trivial
magma: MordellWeilGroupGenus2(Jacobian(C));
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 6.787071 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 6.787071 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(3\) | \(6\) | \(6\) | \(1\) | \(1\) |  |
| \(31\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |  |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
| Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
|---|---|---|
| \(2\) | 2.120.2 | no |
| \(3\) | 3.72.2 | no |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);