Properties

Label 961.a.923521.1
Conductor 961
Discriminant 923521
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

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This is a model for the modular curve $X_0(31)$.

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, -2, 3, 4, -5], R![1, 0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, -2, 3, 4, -5]), R([1, 0, 1, 1]))

$y^2 + (x^3 + x^2 + 1)y = -5x^4 + 4x^3 + 3x^2 - 2x - 3$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 961 \)  =  \( 31^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(923521\)  =  \( 31^{4} \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];

G2 invariants

magma: G2Invariants(C);

\( I_2 \)  =  \(8200\)  =  \( 2^{3} \cdot 5^{2} \cdot 41 \)
\( I_4 \)  =  \(315844\)  =  \( 2^{2} \cdot 281^{2} \)
\( I_6 \)  =  \(753213512\)  =  \( 2^{3} \cdot 94151689 \)
\( I_{10} \)  =  \(3782742016\)  =  \( 2^{12} \cdot 31^{4} \)
\( J_2 \)  =  \(1025\)  =  \( 5^{2} \cdot 41 \)
\( J_4 \)  =  \(40486\)  =  \( 2 \cdot 31 \cdot 653 \)
\( J_6 \)  =  \(2121888\)  =  \( 2^{5} \cdot 3 \cdot 23 \cdot 31^{2} \)
\( J_8 \)  =  \(133954751\)  =  \( 7 \cdot 31^{2} \cdot 19913 \)
\( J_{10} \)  =  \(923521\)  =  \( 31^{4} \)
\( g_1 \)  =  \(1131408212890625/923521\)
\( g_2 \)  =  \(1406419156250/29791\)
\( g_3 \)  =  \(2319780000/961\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(C_2 \) (GAP id : [2,1])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(C_2 \) (GAP id : [2,1])

Rational points

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);

This curve is locally solvable everywhere.

magma: [C![1,-1,0],C![1,0,0]];

All rational points: (1 : -1 : 0), (1 : 0 : 0)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);

2-Selmer rank: \(0\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 5 (p = 31)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: \(\Z/{5}\Z\)

2-torsion field: 3.1.31.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition

Simple over \(\overline{\Q}\)

Endomorphisms

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{5}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).