Minimal equation
Minimal equation
Simplified equation
| $y^2 + xy = -13x^6 - 12x^5 + 63x^4 + 23x^3 - 81x^2 + 27x - 6$ | (homogenize, simplify) |
| $y^2 + xz^2y = -13x^6 - 12x^5z + 63x^4z^2 + 23x^3z^3 - 81x^2z^4 + 27xz^5 - 6z^6$ | (dehomogenize, simplify) |
| $y^2 = -52x^6 - 48x^5 + 252x^4 + 92x^3 - 323x^2 + 108x - 24$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, 27, -81, 23, 63, -12, -13]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, 27, -81, 23, 63, -12, -13], R![0, 1]);
sage: X = HyperellipticCurve(R([-24, 108, -323, 92, 252, -48, -52]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(293712\) | \(=\) | \( 2^{4} \cdot 3 \cdot 29 \cdot 211 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-293712\) | \(=\) | \( - 2^{4} \cdot 3 \cdot 29 \cdot 211 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(211560\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 41 \cdot 43 \) |
| \( I_4 \) | \(=\) | \(4586651160\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 67 \cdot 6269 \) |
| \( I_6 \) | \(=\) | \(198747415326036\) | \(=\) | \( 2^{2} \cdot 3 \cdot 859 \cdot 10211 \cdot 1888247 \) |
| \( I_{10} \) | \(=\) | \(-1174848\) | \(=\) | \( - 2^{6} \cdot 3 \cdot 29 \cdot 211 \) |
| \( J_2 \) | \(=\) | \(105780\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 41 \cdot 43 \) |
| \( J_4 \) | \(=\) | \(-298216510\) | \(=\) | \( - 2 \cdot 5 \cdot 83 \cdot 359297 \) |
| \( J_6 \) | \(=\) | \(3118654812496\) | \(=\) | \( 2^{4} \cdot 17 \cdot 53 \cdot 59 \cdot 3666659 \) |
| \( J_8 \) | \(=\) | \(60239554807311695\) | \(=\) | \( 5 \cdot 12047910961462339 \) |
| \( J_{10} \) | \(=\) | \(-293712\) | \(=\) | \( - 2^{4} \cdot 3 \cdot 29 \cdot 211 \) |
| \( g_1 \) | \(=\) | \(-275915803478661696600000/6119\) | ||
| \( g_2 \) | \(=\) | \(7353619157135452365000/6119\) | ||
| \( g_3 \) | \(=\) | \(-726997965742565986800/6119\) |
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: \(\Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(13x^2 + 12xz - 24z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(13x^2 + 12xz - 24z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(13x^2 + 12xz - 24z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
2-torsion field: 6.2.117268848252.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(3\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.211647 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 0.846589 \) |
| Analytic order of Ш: | \( 16 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(4\) | \(4\) | \(1\) | \(1 - T + 2 T^{2}\) | |
| \(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T^{2} )\) |  |
| \(29\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 10 T + 29 T^{2} )\) | |
| \(211\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 4 T + 211 T^{2} )\) | |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.
| Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
|---|---|---|
| \(2\) | 2.15.1 | yes |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
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);