Properties

Label 1042.a.1042.1
Conductor 1042
Discriminant 1042
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![2, 2, -1, -1, -1], R![0, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 2, -1, -1, -1]), R([0, 1, 0, 1]))

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 1042 \)  =  \( 2 \cdot 521 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(1042\)  =  \( 2 \cdot 521 \)

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);

\( I_2 \)  =  \(-1920\)  =  \( -1 \cdot 2^{7} \cdot 3 \cdot 5 \)
\( I_4 \)  =  \(62592\)  =  \( 2^{7} \cdot 3 \cdot 163 \)
\( I_6 \)  =  \(-46648896\)  =  \( -1 \cdot 2^{6} \cdot 3 \cdot 7 \cdot 61 \cdot 569 \)
\( I_{10} \)  =  \(4268032\)  =  \( 2^{13} \cdot 521 \)
\( J_2 \)  =  \(-240\)  =  \( -1 \cdot 2^{4} \cdot 3 \cdot 5 \)
\( J_4 \)  =  \(1748\)  =  \( 2^{2} \cdot 19 \cdot 23 \)
\( J_6 \)  =  \(5521\)  =  \( 5521 \)
\( J_8 \)  =  \(-1095136\)  =  \( -1 \cdot 2^{5} \cdot 7 \cdot 4889 \)
\( J_{10} \)  =  \(1042\)  =  \( 2 \cdot 521 \)
\( g_1 \)  =  \(-398131200000/521\)
\( g_2 \)  =  \(-12082176000/521\)
\( g_3 \)  =  \(159004800/521\)
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![-1,1,1],C![1,-1,0],C![1,0,0]];

All rational points: (-1 : 1 : 1), (1 : -1 : 0), (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 = 521)

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

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

2-torsion field: 5.1.16672.1

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 \(\Q\).