Properties

Label 323375.a.323375.1
Conductor $323375$
Discriminant $-323375$
Mordell-Weil group \(\Z/{2}\Z\)
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

$y^2 + xy = -5x^6 - 5x^5 + 4x^4 - x^3 + 11x^2 + 4x - 9$ (homogenize, simplify)
$y^2 + xz^2y = -5x^6 - 5x^5z + 4x^4z^2 - x^3z^3 + 11x^2z^4 + 4xz^5 - 9z^6$ (dehomogenize, simplify)
$y^2 = -20x^6 - 20x^5 + 16x^4 - 4x^3 + 45x^2 + 16x - 36$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-9, 4, 11, -1, 4, -5, -5]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-9, 4, 11, -1, 4, -5, -5], R![0, 1]);
 
sage: X = HyperellipticCurve(R([-36, 16, 45, -4, 16, -20, -20]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(323375\) \(=\) \( 5^{3} \cdot 13 \cdot 199 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-323375\) \(=\) \( - 5^{3} \cdot 13 \cdot 199 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(49256\) \(=\)  \( 2^{3} \cdot 47 \cdot 131 \)
\( I_4 \)  \(=\) \(23881480\) \(=\)  \( 2^{3} \cdot 5 \cdot 7 \cdot 19 \cdot 67^{2} \)
\( I_6 \)  \(=\) \(574290197005\) \(=\)  \( 5 \cdot 7 \cdot 197 \cdot 83290819 \)
\( I_{10} \)  \(=\) \(1293500\) \(=\)  \( 2^{2} \cdot 5^{3} \cdot 13 \cdot 199 \)
\( J_2 \)  \(=\) \(24628\) \(=\)  \( 2^{2} \cdot 47 \cdot 131 \)
\( J_4 \)  \(=\) \(21292186\) \(=\)  \( 2 \cdot 10646093 \)
\( J_6 \)  \(=\) \(-2002408209\) \(=\)  \( - 3^{5} \cdot 8240363 \)
\( J_8 \)  \(=\) \(-125668123507462\) \(=\)  \( - 2 \cdot 62834061753731 \)
\( J_{10} \)  \(=\) \(323375\) \(=\)  \( 5^{3} \cdot 13 \cdot 199 \)
\( g_1 \)  \(=\) \(9060365643842583098368/323375\)
\( g_2 \)  \(=\) \(318058997757850118272/323375\)
\( g_3 \)  \(=\) \(-1214537439195194256/323375\)

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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

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: \(\Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(5x^2 - 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(5x^2 - 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(5x^2 - 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)

2-torsion field: 6.2.1338513800000.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 0.687405 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.749622 \)
Analytic order of Ш: \( 16 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(5\) \(3\) \(3\) \(1\) \(1 - T\)
\(13\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 13 T^{2} )\)
\(199\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 16 T + 199 T^{2} )\)

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?
\(2\) 2.15.1 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);