Properties

Label 708.a.19116.1
Conductor $708$
Discriminant $-19116$
Mordell-Weil group \(\Z/{10}\Z\)
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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 4, 4, 0, 0, -1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 4, 4, 0, 0, -1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-3, 16, 16, 2, 0, -4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(708\) \(=\) \( 2^{2} \cdot 3 \cdot 59 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-19116\) \(=\) \( - 2^{2} \cdot 3^{4} \cdot 59 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(908\) \(=\)  \( 2^{2} \cdot 227 \)
\( I_4 \)  \(=\) \(-132815\) \(=\)  \( - 5 \cdot 101 \cdot 263 \)
\( I_6 \)  \(=\) \(8426215\) \(=\)  \( 5 \cdot 7 \cdot 19 \cdot 12671 \)
\( I_{10} \)  \(=\) \(2446848\) \(=\)  \( 2^{9} \cdot 3^{4} \cdot 59 \)
\( J_2 \)  \(=\) \(227\) \(=\)  \( 227 \)
\( J_4 \)  \(=\) \(7681\) \(=\)  \( 7681 \)
\( J_6 \)  \(=\) \(-438901\) \(=\)  \( - 43 \cdot 59 \cdot 173 \)
\( J_8 \)  \(=\) \(-39657072\) \(=\)  \( - 2^{4} \cdot 3 \cdot 7^{2} \cdot 13 \cdot 1297 \)
\( J_{10} \)  \(=\) \(19116\) \(=\)  \( 2^{2} \cdot 3^{4} \cdot 59 \)
\( g_1 \)  \(=\) \(602738989907/19116\)
\( g_2 \)  \(=\) \(89845294523/19116\)
\( g_3 \)  \(=\) \(-383324231/324\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (3 : -14 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (3 : -14 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : 0 : 1),\, (3 : 0 : 1)\)

magma: [C![-1,0,1],C![1,-1,0],C![1,0,0],C![3,-14,1]]; // minimal model
 
magma: [C![-1,0,1],C![1,-1,0],C![1,1,0],C![3,0,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(10\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(10\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0\) \(10\)

2-torsion field: 4.2.3776.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 16.26718 \)
Tamagawa product: \( 2 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.325343 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(2\) \(1\) \(1 + T^{2}\)
\(3\) \(1\) \(4\) \(2\) \(( 1 + T )( 1 + T + 3 T^{2} )\)
\(59\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 59 T^{2} )\)

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?
\(2\) 2.30.3 yes
\(5\) not computed 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);