Properties

Label 388.a.776.1
Conductor 388
Discriminant 776
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![0, 1, 2, 0, -1], R![1, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 2, 0, -1]), R([1, 1, 0, 1]))

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 388 \)  =  \( 2^{2} \cdot 97 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(776\)  =  \( 2^{3} \cdot 97 \)

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);

\( I_2 \)  =  \(72\)  =  \( 2^{3} \cdot 3^{2} \)
\( I_4 \)  =  \(6276\)  =  \( 2^{2} \cdot 3 \cdot 523 \)
\( I_6 \)  =  \(-109944\)  =  \( -1 \cdot 2^{3} \cdot 3^{3} \cdot 509 \)
\( I_{10} \)  =  \(3178496\)  =  \( 2^{15} \cdot 97 \)
\( J_2 \)  =  \(9\)  =  \( 3^{2} \)
\( J_4 \)  =  \(-62\)  =  \( -1 \cdot 2 \cdot 31 \)
\( J_6 \)  =  \(356\)  =  \( 2^{2} \cdot 89 \)
\( J_8 \)  =  \(-160\)  =  \( -1 \cdot 2^{5} \cdot 5 \)
\( J_{10} \)  =  \(776\)  =  \( 2^{3} \cdot 97 \)
\( g_1 \)  =  \(59049/776\)
\( g_2 \)  =  \(-22599/388\)
\( g_3 \)  =  \(7209/194\)
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![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];

All rational points: (-1 : 0 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (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: 3 (p = 2), 1 (p = 97)

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

Torsion: \(\Z/{21}\Z\)

2-torsion field: 6.2.49664.1

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 \(\Q\).