↑

Properties

Label 713293.a.713293.1
Conductor $713293$
Discriminant $-713293$
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

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = -x^5 + x^4 - 4x^2 + 7x - 2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -x^5z + x^4z^2 - 4x^2z^4 + 7xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 4x^5 + 4x^4 + 2x^3 - 16x^2 + 28x - 7$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 7, -4, 0, 1, -1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 7, -4, 0, 1, -1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-7, 28, -16, 2, 4, -4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(713293\) \(=\) \( 7^{2} \cdot 14557 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-713293\) \(=\) \( - 7^{2} \cdot 14557 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(876\) \(=\)  \( 2^{2} \cdot 3 \cdot 73 \)
\( I_4 \)  \(=\) \(-39591\) \(=\)  \( - 3^{2} \cdot 53 \cdot 83 \)
\( I_6 \)  \(=\) \(-3724893\) \(=\)  \( - 3^{3} \cdot 19 \cdot 53 \cdot 137 \)
\( I_{10} \)  \(=\) \(91301504\) \(=\)  \( 2^{7} \cdot 7^{2} \cdot 14557 \)
\( J_2 \)  \(=\) \(219\) \(=\)  \( 3 \cdot 73 \)
\( J_4 \)  \(=\) \(3648\) \(=\)  \( 2^{6} \cdot 3 \cdot 19 \)
\( J_6 \)  \(=\) \(-24304\) \(=\)  \( - 2^{4} \cdot 7^{2} \cdot 31 \)
\( J_8 \)  \(=\) \(-4657620\) \(=\)  \( - 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 7057 \)
\( J_{10} \)  \(=\) \(713293\) \(=\)  \( 7^{2} \cdot 14557 \)
\( g_1 \)  \(=\) \(503756397099/713293\)
\( g_2 \)  \(=\) \(38316618432/713293\)
\( g_3 \)  \(=\) \(-23788656/14557\)

  (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

Known points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -3 : 3),\, (2 : -4 : 1),\, (2 : -5 : 1),\, (1 : -25 : 3)\)
Known points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -3 : 3),\, (2 : -4 : 1),\, (2 : -5 : 1),\, (1 : -25 : 3)\)
Known points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (2 : -1 : 1),\, (2 : 1 : 1),\, (1 : -22 : 3),\, (1 : 22 : 3)\)

magma: [C![1,-25,3],C![1,-3,3],C![1,-1,0],C![1,0,0],C![2,-5,1],C![2,-4,1]]; // minimal model
 
magma: [C![1,-22,3],C![1,22,3],C![1,-1,0],C![1,1,0],C![2,-1,1],C![2,1,1]]; // 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
\((2 : -5 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-5z^3\) \(0.424630\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(-x^2z\) \(0.449837\) \(\infty\)
Generator $D_0$ Height Order
\((2 : -5 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-5z^3\) \(0.424630\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(-x^2z\) \(0.449837\) \(\infty\)
Generator $D_0$ Height Order
\((2 : -1 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 9z^3\) \(0.424630\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2x^2z + z^3\) \(0.449837\) \(\infty\)

2-torsion field: 6.4.45650752.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.188505 \)
Real period: \( 12.82609 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 2.417784 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(7\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)
\(14557\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 203 T + 14557 T^{2} )\)

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);