Properties

Label 12890.a.25780.1
Conductor $12890$
Discriminant $25780$
Mordell-Weil group \(\Z \oplus \Z\)
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

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

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

Invariants

Conductor: \( N \)  \(=\)  \(12890\) \(=\) \( 2 \cdot 5 \cdot 1289 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(25780\) \(=\) \( 2^{2} \cdot 5 \cdot 1289 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(136\) \(=\)  \( 2^{3} \cdot 17 \)
\( I_4 \)  \(=\) \(-392\) \(=\)  \( - 2^{3} \cdot 7^{2} \)
\( I_6 \)  \(=\) \(32749\) \(=\)  \( 32749 \)
\( I_{10} \)  \(=\) \(-103120\) \(=\)  \( - 2^{4} \cdot 5 \cdot 1289 \)
\( J_2 \)  \(=\) \(68\) \(=\)  \( 2^{2} \cdot 17 \)
\( J_4 \)  \(=\) \(258\) \(=\)  \( 2 \cdot 3 \cdot 43 \)
\( J_6 \)  \(=\) \(-4145\) \(=\)  \( - 5 \cdot 829 \)
\( J_8 \)  \(=\) \(-87106\) \(=\)  \( - 2 \cdot 97 \cdot 449 \)
\( J_{10} \)  \(=\) \(-25780\) \(=\)  \( - 2^{2} \cdot 5 \cdot 1289 \)
\( g_1 \)  \(=\) \(-363483392/6445\)
\( g_2 \)  \(=\) \(-20280864/6445\)
\( g_3 \)  \(=\) \(958324/1289\)

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

Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((1 : -2 : 1)\) \((-1 : -4 : 3)\) \((-1 : -14 : 3)\) \((-6 : -99 : 5)\) \((-6 : 124 : 5)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((1 : -2 : 1)\) \((-1 : -4 : 3)\) \((-1 : -14 : 3)\) \((-6 : -99 : 5)\) \((-6 : 124 : 5)\)
Known points
\((1 : -2 : 0)\) \((1 : 2 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -2 : 1)\)
\((1 : 2 : 1)\) \((-1 : -10 : 3)\) \((-1 : 10 : 3)\) \((-6 : -223 : 5)\) \((-6 : 223 : 5)\)

magma: [C![-6,-99,5],C![-6,124,5],C![-1,-14,3],C![-1,-4,3],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,1],C![1,1,0]]; // minimal model
 
magma: [C![-6,-223,5],C![-6,223,5],C![-1,-10,3],C![-1,10,3],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-2,1],C![1,-2,0],C![1,2,1],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: \(\Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0.333121\) \(\infty\)
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.047337\) \(\infty\)
Generator $D_0$ Height Order
\((0 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0.333121\) \(\infty\)
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.047337\) \(\infty\)
Generator $D_0$ Height Order
\((0 : 1 : 1) + (1 : -2 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 + z^3\) \(0.333121\) \(\infty\)
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.047337\) \(\infty\)

2-torsion field: 5.1.103120.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.015716 \)
Real period: \( 18.32866 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.576126 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(1289\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 6 T + 1289 T^{2} )\)

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);