Properties

Label 614400.a.614400.1
Conductor $614400$
Discriminant $614400$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
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

$y^2 = 6x^6 - 16x^5 - x^4 + 23x^3 - 11x - 3$ (homogenize, simplify)
$y^2 = 6x^6 - 16x^5z - x^4z^2 + 23x^3z^3 - 11xz^5 - 3z^6$ (dehomogenize, simplify)
$y^2 = 6x^6 - 16x^5 - x^4 + 23x^3 - 11x - 3$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, -11, 0, 23, -1, -16, 6]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, -11, 0, 23, -1, -16, 6], R![]);
 
sage: X = HyperellipticCurve(R([-3, -11, 0, 23, -1, -16, 6]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(614400\) \(=\) \( 2^{13} \cdot 3 \cdot 5^{2} \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(614400,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(614400\) \(=\) \( 2^{13} \cdot 3 \cdot 5^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(7267\) \(=\)  \( 13^{2} \cdot 43 \)
\( I_4 \)  \(=\) \(2059\) \(=\)  \( 29 \cdot 71 \)
\( I_6 \)  \(=\) \(5009595\) \(=\)  \( 3 \cdot 5 \cdot 333973 \)
\( I_{10} \)  \(=\) \(75\) \(=\)  \( 3 \cdot 5^{2} \)
\( J_2 \)  \(=\) \(29068\) \(=\)  \( 2^{2} \cdot 13^{2} \cdot 43 \)
\( J_4 \)  \(=\) \(35184230\) \(=\)  \( 2 \cdot 5 \cdot 79 \cdot 44537 \)
\( J_6 \)  \(=\) \(56746435396\) \(=\)  \( 2^{2} \cdot 7 \cdot 191 \cdot 2399 \cdot 4423 \)
\( J_8 \)  \(=\) \(102893835849507\) \(=\)  \( 3^{5} \cdot 41 \cdot 89 \cdot 116040401 \)
\( J_{10} \)  \(=\) \(614400\) \(=\)  \( 2^{13} \cdot 3 \cdot 5^{2} \)
\( g_1 \)  \(=\) \(20266362241049681107/600\)
\( g_2 \)  \(=\) \(1350247965566071949/960\)
\( g_3 \)  \(=\) \(749184726636798361/9600\)

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

No rational points are known for this curve.
No rational points are known for this curve.
No rational points are known for this curve.

magma: []; // minimal model
 
magma: []; // 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: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(4xz^2 + 2z^3\) \(1.844732\) \(\infty\)
\(D_0 - D_\infty\) \(2x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - D_\infty\) \(3x^2 - 2xz - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(4xz^2 + 2z^3\) \(1.844732\) \(\infty\)
\(D_0 - D_\infty\) \(2x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - D_\infty\) \(3x^2 - 2xz - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2 + z^3\) \(1.844732\) \(\infty\)
\(D_0 - D_\infty\) \(2x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - D_\infty\) \(3x^2 - 2xz - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 8.8.3317760000.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(5\)
Regulator: \( 1.844732 \)
Real period: \( 7.605509 \)
Tamagawa product: \( 1 \)
Torsion order:\( 4 \)
Leading coefficient: \( 3.507533 \)
Analytic order of Ш: \( 4 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(13\) \(13\) \(1\) \(1\)
\(3\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(5\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

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?
\(2\) 2.90.6 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);