Properties

Label 417152.a.417152.1
Conductor $417152$
Discriminant $-417152$
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

$y^2 + (x^3 + x)y = -x^6 - 7x^5 - 15x^4 + 12x^3 + 16x^2 + x - 9$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 - 7x^5z - 15x^4z^2 + 12x^3z^3 + 16x^2z^4 + xz^5 - 9z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 - 28x^5 - 58x^4 + 48x^3 + 65x^2 + 4x - 36$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-9, 1, 16, 12, -15, -7, -1]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-9, 1, 16, 12, -15, -7, -1], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-36, 4, 65, 48, -58, -28, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(417152\) \(=\) \( 2^{7} \cdot 3259 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-417152\) \(=\) \( - 2^{7} \cdot 3259 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(5468\) \(=\)  \( 2^{2} \cdot 1367 \)
\( I_4 \)  \(=\) \(10622470\) \(=\)  \( 2 \cdot 5 \cdot 47 \cdot 97 \cdot 233 \)
\( I_6 \)  \(=\) \(7817476501\) \(=\)  \( 7817476501 \)
\( I_{10} \)  \(=\) \(-52144\) \(=\)  \( - 2^{4} \cdot 3259 \)
\( J_2 \)  \(=\) \(5468\) \(=\)  \( 2^{2} \cdot 1367 \)
\( J_4 \)  \(=\) \(-5835854\) \(=\)  \( - 2 \cdot 2917927 \)
\( J_6 \)  \(=\) \(4185810564\) \(=\)  \( 2^{2} \cdot 3 \cdot 13 \cdot 439 \cdot 61121 \)
\( J_8 \)  \(=\) \(-2792294936341\) \(=\)  \( - 23 \cdot 121404127667 \)
\( J_{10} \)  \(=\) \(-417152\) \(=\)  \( - 2^{7} \cdot 3259 \)
\( g_1 \)  \(=\) \(-38188496456892856/3259\)
\( g_2 \)  \(=\) \(7453838285890001/3259\)
\( g_3 \)  \(=\) \(-1955494539257649/6518\)

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 except over $\Q_{2}$.

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
\(D_0 - D_\infty\) \(x^2 + 4xz - 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-21xz^2 + 16z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 4xz - 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-21xz^2 + 16z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 4xz - 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(x^3 - 41xz^2 + 32z^3\) \(0\) \(2\)

2-torsion field: 6.2.21751973888.4

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 0.691503 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.766013 \)
Analytic order of Ш: \( 16 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(7\) \(7\) \(1\) \(1 - T\)
\(3259\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 84 T + 3259 T^{2} )\)

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);