Properties

Label 862.b.862.1
Conductor 862
Discriminant 862
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Show commands for: Magma / SageMath

Minimal equation

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

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 862 \)  =  \( 2 \cdot 431 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(862\)  =  \( 2 \cdot 431 \)

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);

\( I_2 \)  =  \(2208\)  =  \( 2^{5} \cdot 3 \cdot 23 \)
\( I_4 \)  =  \(11136\)  =  \( 2^{7} \cdot 3 \cdot 29 \)
\( I_6 \)  =  \(7216320\)  =  \( 2^{6} \cdot 3 \cdot 5 \cdot 7517 \)
\( I_{10} \)  =  \(3530752\)  =  \( 2^{13} \cdot 431 \)
\( J_2 \)  =  \(276\)  =  \( 2^{2} \cdot 3 \cdot 23 \)
\( J_4 \)  =  \(3058\)  =  \( 2 \cdot 11 \cdot 139 \)
\( J_6 \)  =  \(45033\)  =  \( 3 \cdot 17 \cdot 883 \)
\( J_8 \)  =  \(769436\)  =  \( 2^{2} \cdot 149 \cdot 1291 \)
\( J_{10} \)  =  \(862\)  =  \( 2 \cdot 431 \)
\( g_1 \)  =  \(800784050688/431\)
\( g_2 \)  =  \(32146576704/431\)
\( g_3 \)  =  \(1715216904/431\)
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,-1,1],C![1,0,0]];

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

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

Number of rational Weierstrass points: \(1\)

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: 1 (p = 2), 1 (p = 431)

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

Torsion: \(\Z/{9}\Z\)

2-torsion field: 5.1.13792.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition

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

Endomorphisms

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 \(\Q\).