Properties

Label 1309.a.9163.2
Conductor $1309$
Discriminant $-9163$
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

$y^2 + x^2y = -x^6 + 20x^5 - 128x^4 + 248x^3 + 32x^2 + x$ (homogenize, simplify)
$y^2 + x^2zy = -x^6 + 20x^5z - 128x^4z^2 + 248x^3z^3 + 32x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = -4x^6 + 80x^5 - 511x^4 + 992x^3 + 128x^2 + 4x$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 32, 248, -128, 20, -1]), R([0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 32, 248, -128, 20, -1], R![0, 0, 1]);
 
sage: X = HyperellipticCurve(R([0, 4, 128, 992, -511, 80, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1309\) \(=\) \( 7 \cdot 11 \cdot 17 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-9163\) \(=\) \( - 7^{2} \cdot 11 \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1740928\) \(=\)  \( 2^{7} \cdot 7 \cdot 29 \cdot 67 \)
\( I_4 \)  \(=\) \(18364336\) \(=\)  \( 2^{4} \cdot 19 \cdot 193 \cdot 313 \)
\( I_6 \)  \(=\) \(10627907866359\) \(=\)  \( 3 \cdot 7 \cdot 79 \cdot 8209 \cdot 780389 \)
\( I_{10} \)  \(=\) \(-36652\) \(=\)  \( - 2^{2} \cdot 7^{2} \cdot 11 \cdot 17 \)
\( J_2 \)  \(=\) \(870464\) \(=\)  \( 2^{6} \cdot 7 \cdot 29 \cdot 67 \)
\( J_4 \)  \(=\) \(31568088248\) \(=\)  \( 2^{3} \cdot 23 \cdot 127 \cdot 1350911 \)
\( J_6 \)  \(=\) \(1526311891463681\) \(=\)  \( 7^{2} \cdot 107 \cdot 291114226867 \)
\( J_8 \)  \(=\) \(83013839664381477120\) \(=\)  \( 2^{8} \cdot 3 \cdot 5 \cdot 17 \cdot 1271658083094079 \)
\( J_{10} \)  \(=\) \(-9163\) \(=\)  \( - 7^{2} \cdot 11 \cdot 17 \)
\( g_1 \)  \(=\) \(-10199009421268235327400574976/187\)
\( g_2 \)  \(=\) \(-424917527486779411910361088/187\)
\( g_3 \)  \(=\) \(-23602001682171372468506624/187\)

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: \((0 : 0 : 1)\)
All points: \((0 : 0 : 1)\)
All points: \((0 : 0 : 1)\)

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

Number of rational Weierstrass points: \(1\)

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
\(D_0 - D_\infty\) \(4x^2 - 16xz - z^2\) \(=\) \(0,\) \(8y\) \(=\) \(-16xz^2 - z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(4x^2 - 16xz - z^2\) \(=\) \(0,\) \(8y\) \(=\) \(-16xz^2 - z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(4x^2 - 16xz - z^2\) \(=\) \(0,\) \(8y\) \(=\) \(x^2z - 32xz^2 - 2z^3\) \(0\) \(2\)

2-torsion field: 6.2.9511568.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 0.846601 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.423300 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(7\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + 7 T^{2} )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 11 T^{2} )\)
\(17\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 6 T + 17 T^{2} )\)

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?
\(2\) 2.60.1 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);