Properties

Label 1038.a.1038.2
Conductor 1038
Discriminant -1038
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, 2, 0, 1], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 1, 2, 0, 1]), R([1, 0, 0, 1]))

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 1038 \)  =  \( 2 \cdot 3 \cdot 173 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-1038\)  =  \( -1 \cdot 2 \cdot 3 \cdot 173 \)

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);

\( I_2 \)  =  \(-1688\)  =  \( -1 \cdot 2^{3} \cdot 211 \)
\( I_4 \)  =  \(16516\)  =  \( 2^{2} \cdot 4129 \)
\( I_6 \)  =  \(-9071864\)  =  \( -1 \cdot 2^{3} \cdot 127 \cdot 8929 \)
\( I_{10} \)  =  \(-4251648\)  =  \( -1 \cdot 2^{13} \cdot 3 \cdot 173 \)
\( J_2 \)  =  \(-211\)  =  \( -1 \cdot 211 \)
\( J_4 \)  =  \(1683\)  =  \( 3^{2} \cdot 11 \cdot 17 \)
\( J_6 \)  =  \(-16079\)  =  \( -1 \cdot 7 \cdot 2297 \)
\( J_8 \)  =  \(140045\)  =  \( 5 \cdot 37 \cdot 757 \)
\( J_{10} \)  =  \(-1038\)  =  \( -1 \cdot 2 \cdot 3 \cdot 173 \)
\( g_1 \)  =  \(418227202051/1038\)
\( g_2 \)  =  \(5269995291/346\)
\( g_3 \)  =  \(715853159/1038\)
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: \(1\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 1 (p = 2), 1 (p = 3), 1 (p = 173)

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

Torsion: \(\Z/{6}\Z\)

2-torsion field: 6.4.51717312.1

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 \(\Q\).