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Properties

Label 7562.a.241984.1
Conductor $7562$
Discriminant $241984$
Mordell-Weil group \(\Z/{9}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + 1)y = x^6 - 3x^5 + 2x^4 - x^3 - x^2 + 3x + 1$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = x^6 - 3x^5z + 2x^4z^2 - x^3z^3 - x^2z^4 + 3xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = 4x^6 - 12x^5 + 9x^4 - 4x^3 - 2x^2 + 12x + 5$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, -1, -1, 2, -3, 1]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, -1, -1, 2, -3, 1], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([5, 12, -2, -4, 9, -12, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(7562\) \(=\) \( 2 \cdot 19 \cdot 199 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(241984\) \(=\) \( 2^{6} \cdot 19 \cdot 199 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2544\) \(=\)  \( 2^{4} \cdot 3 \cdot 53 \)
\( I_4 \)  \(=\) \(102576\) \(=\)  \( 2^{4} \cdot 3 \cdot 2137 \)
\( I_6 \)  \(=\) \(97720377\) \(=\)  \( 3 \cdot 32573459 \)
\( I_{10} \)  \(=\) \(-967936\) \(=\)  \( - 2^{8} \cdot 19 \cdot 199 \)
\( J_2 \)  \(=\) \(1272\) \(=\)  \( 2^{3} \cdot 3 \cdot 53 \)
\( J_4 \)  \(=\) \(50320\) \(=\)  \( 2^{4} \cdot 5 \cdot 17 \cdot 37 \)
\( J_6 \)  \(=\) \(-53169\) \(=\)  \( - 3 \cdot 37 \cdot 479 \)
\( J_8 \)  \(=\) \(-649933342\) \(=\)  \( - 2 \cdot 19 \cdot 37 \cdot 503 \cdot 919 \)
\( J_{10} \)  \(=\) \(-241984\) \(=\)  \( - 2^{6} \cdot 19 \cdot 199 \)
\( g_1 \)  \(=\) \(-52030210457088/3781\)
\( g_2 \)  \(=\) \(-1618161978240/3781\)
\( g_3 \)  \(=\) \(1344165489/3781\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : -5 : 2)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : -5 : 2)\)
All points: \((1 : -2 : 0),\, (1 : 2 : 0),\, (-1 : 0 : 2)\)

magma: [C![-1,-5,2],C![1,-1,0],C![1,1,0]]; // minimal model
 
magma: [C![-1,0,2],C![1,-2,0],C![1,2,0]]; // 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: \(\Z/{9}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2x^2z\) \(0\) \(9\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2x^2z\) \(0\) \(9\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -2 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(2x^3 - 3x^2z + z^3\) \(0\) \(9\)

2-torsion field: 5.1.60496.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 17.89869 \)
Tamagawa product: \( 6 \)
Torsion order:\( 9 \)
Leading coefficient: \( 1.325829 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(6\) \(6\) \(( 1 - T )( 1 + 2 T^{2} )\)
\(19\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 7 T + 19 T^{2} )\)
\(199\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 16 T + 199 T^{2} )\)

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
\(3\) 3.80.1 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);