Properties

Label 13696.b.109568.1
Conductor $13696$
Discriminant $-109568$
Mordell-Weil group \(\Z/{2}\Z\)
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

$y^2 + x^2y = x^5 + 12x^4 - 2x^3 - 253x^2 + 288x - 83$ (homogenize, simplify)
$y^2 + x^2zy = x^5z + 12x^4z^2 - 2x^3z^3 - 253x^2z^4 + 288xz^5 - 83z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 + 49x^4 - 8x^3 - 1012x^2 + 1152x - 332$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-83, 288, -253, -2, 12, 1]), R([0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-83, 288, -253, -2, 12, 1], R![0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-332, 1152, -1012, -8, 49, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(13696\) \(=\) \( 2^{7} \cdot 107 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-109568\) \(=\) \( - 2^{10} \cdot 107 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(122264\) \(=\)  \( 2^{3} \cdot 17 \cdot 29 \cdot 31 \)
\( I_4 \)  \(=\) \(117761944\) \(=\)  \( 2^{3} \cdot 14720243 \)
\( I_6 \)  \(=\) \(4373366507708\) \(=\)  \( 2^{2} \cdot 11 \cdot 2311 \cdot 5569 \cdot 7723 \)
\( I_{10} \)  \(=\) \(-13696\) \(=\)  \( - 2^{7} \cdot 107 \)
\( J_2 \)  \(=\) \(122264\) \(=\)  \( 2^{3} \cdot 17 \cdot 29 \cdot 31 \)
\( J_4 \)  \(=\) \(544345608\) \(=\)  \( 2^{3} \cdot 3 \cdot 22681067 \)
\( J_6 \)  \(=\) \(3009566254336\) \(=\)  \( 2^{8} \cdot 239 \cdot 49188779 \)
\( J_8 \)  \(=\) \(17912366892811760\) \(=\)  \( 2^{4} \cdot 5 \cdot 181 \cdot 197 \cdot 4231 \cdot 1484141 \)
\( J_{10} \)  \(=\) \(-109568\) \(=\)  \( - 2^{10} \cdot 107 \)
\( g_1 \)  \(=\) \(-26680443465746439572576/107\)
\( g_2 \)  \(=\) \(-971562104378078994348/107\)
\( g_3 \)  \(=\) \(-43934041117291009744/107\)

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: \((1 : 0 : 0)\)
All points: \((1 : 0 : 0)\)
All points: \((1 : 0 : 0)\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 + z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 + z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - 4xz^2 + 2z^3\) \(0\) \(2\)

2-torsion field: 6.2.5861888.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 0.378972 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 1.705377 \)
Analytic order of Ш: \( 9 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(7\) \(10\) \(2\) \(1 - T\)
\(107\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 107 T^{2} )\)

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?
\(2\) 2.60.1 yes
\(3\) 3.80.2 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);