Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x + 1)y = -x^6 - 10x^5 - 23x^4 + 34x^3 + 19x^2 - 15x - 7$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2 + z^3)y = -x^6 - 10x^5z - 23x^4z^2 + 34x^3z^3 + 19x^2z^4 - 15xz^5 - 7z^6$ | (dehomogenize, simplify) |
| $y^2 = -4x^6 - 40x^5 - 91x^4 + 138x^3 + 79x^2 - 58x - 27$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-7, -15, 19, 34, -23, -10, -1]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-7, -15, 19, 34, -23, -10, -1], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([-27, -58, 79, 138, -91, -40, -4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(342871\) | \(=\) | \( 342871 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-342871\) | \(=\) | \( -342871 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(148084\) | \(=\) | \( 2^{2} \cdot 37021 \) |
| \( I_4 \) | \(=\) | \(1415381689\) | \(=\) | \( 1861 \cdot 760549 \) |
| \( I_6 \) | \(=\) | \(52115503116749\) | \(=\) | \( 17 \cdot 19 \cdot 1109 \cdot 2969 \cdot 49003 \) |
| \( I_{10} \) | \(=\) | \(-43887488\) | \(=\) | \( - 2^{7} \cdot 342871 \) |
| \( J_2 \) | \(=\) | \(37021\) | \(=\) | \( 37021 \) |
| \( J_4 \) | \(=\) | \(-1867802\) | \(=\) | \( - 2 \cdot 103 \cdot 9067 \) |
| \( J_6 \) | \(=\) | \(93761116\) | \(=\) | \( 2^{2} \cdot 83 \cdot 282413 \) |
| \( J_8 \) | \(=\) | \(-4388508942\) | \(=\) | \( - 2 \cdot 3 \cdot 731418157 \) |
| \( J_{10} \) | \(=\) | \(-342871\) | \(=\) | \( -342871 \) |
| \( g_1 \) | \(=\) | \(-69540967411549073069101/342871\) | ||
| \( g_2 \) | \(=\) | \(94770958473167416322/342871\) | ||
| \( g_3 \) | \(=\) | \(-128504713926916156/342871\) |
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
All points: \((-1 : -3 : 2)\)
magma: [C![-1,-3,2]]; // minimal model
magma: [C![-1,0,2]]; // simplified model
Number of rational Weierstrass points: \(1\)
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));
2-torsion field: 5.3.5485936.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.407072 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 3.663648 \) |
| Analytic order of Ш: | \( 9 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(342871\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 710 T + 342871 T^{2} )\) |  |
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? |
|---|---|---|
| \(2\) | 2.6.1 | no |
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);