Properties

Label 13006.b.832384.1
Conductor $13006$
Discriminant $-832384$
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

$y^2 + (x^3 + x)y = -2x^4 + 4x^2 - 3x + 1$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -2x^4z^2 + 4x^2z^4 - 3xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 6x^4 + 17x^2 - 12x + 4$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -3, 4, 0, -2]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -3, 4, 0, -2], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([4, -12, 17, 0, -6, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(13006\) \(=\) \( 2 \cdot 7 \cdot 929 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-832384\) \(=\) \( - 2^{7} \cdot 7 \cdot 929 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(168\) \(=\)  \( 2^{3} \cdot 3 \cdot 7 \)
\( I_4 \)  \(=\) \(13020\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 31 \)
\( I_6 \)  \(=\) \(467967\) \(=\)  \( 3 \cdot 389 \cdot 401 \)
\( I_{10} \)  \(=\) \(-3329536\) \(=\)  \( - 2^{9} \cdot 7 \cdot 929 \)
\( J_2 \)  \(=\) \(84\) \(=\)  \( 2^{2} \cdot 3 \cdot 7 \)
\( J_4 \)  \(=\) \(-1876\) \(=\)  \( - 2^{2} \cdot 7 \cdot 67 \)
\( J_6 \)  \(=\) \(9\) \(=\)  \( 3^{2} \)
\( J_8 \)  \(=\) \(-879655\) \(=\)  \( - 5 \cdot 7 \cdot 41 \cdot 613 \)
\( J_{10} \)  \(=\) \(-832384\) \(=\)  \( - 2^{7} \cdot 7 \cdot 929 \)
\( g_1 \)  \(=\) \(-4667544/929\)
\( g_2 \)  \(=\) \(1240974/929\)
\( g_3 \)  \(=\) \(-567/7432\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((-2 : 1 : 1)\)
\((1 : -2 : 1)\) \((1 : 3 : 2)\) \((2 : -3 : 1)\) \((2 : -7 : 1)\) \((1 : -8 : 2)\) \((-2 : 9 : 1)\)
\((-2 : -45 : 3)\) \((-2 : 71 : 3)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((-2 : 1 : 1)\)
\((1 : -2 : 1)\) \((1 : 3 : 2)\) \((2 : -3 : 1)\) \((2 : -7 : 1)\) \((1 : -8 : 2)\) \((-2 : 9 : 1)\)
\((-2 : -45 : 3)\) \((-2 : 71 : 3)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -2 : 1)\) \((0 : 2 : 1)\) \((1 : -2 : 1)\) \((1 : 2 : 1)\)
\((2 : -4 : 1)\) \((2 : 4 : 1)\) \((-2 : -8 : 1)\) \((-2 : 8 : 1)\) \((1 : -11 : 2)\) \((1 : 11 : 2)\)
\((-2 : -116 : 3)\) \((-2 : 116 : 3)\)

magma: [C![-2,-45,3],C![-2,1,1],C![-2,9,1],C![-2,71,3],C![0,-1,1],C![0,1,1],C![1,-8,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![1,3,2],C![2,-7,1],C![2,-3,1]]; // minimal model
 
magma: [C![-2,-116,3],C![-2,-8,1],C![-2,8,1],C![-2,116,3],C![0,-2,1],C![0,2,1],C![1,-11,2],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![1,11,2],C![2,-4,1],C![2,4,1]]; // 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: \(\Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.582012\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.008777\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.582012\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.008777\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -2 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - xz^2 - 2z^3\) \(0.582012\) \(\infty\)
\((0 : -2 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + xz^2 - 2z^3\) \(0.008777\) \(\infty\)

2-torsion field: 6.0.832384.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.005091 \)
Real period: \( 15.77762 \)
Tamagawa product: \( 7 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.562268 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(7\) \(7\) \(( 1 - T )( 1 + 2 T + 2 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 5 T + 7 T^{2} )\)
\(929\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 9 T + 929 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .

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);