Properties

Label 100130.a.200260.1
Conductor 100130
Discriminant -200260
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, 1, 0, 6, 5, 1], R![1, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, 0, 6, 5, 1]), R([1, 1, 1]))

$y^2 + (x^2 + x + 1)y = x^6 + 5x^5 + 6x^4 + x^2 + x$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 100130 \)  =  \( 2 \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-200260\)  =  \( -1 \cdot 2^{2} \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)

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);

\( I_2 \)  =  \(1064\)  =  \( 2^{3} \cdot 7 \cdot 19 \)
\( I_4 \)  =  \(872740\)  =  \( 2^{2} \cdot 5 \cdot 11 \cdot 3967 \)
\( I_6 \)  =  \(-8682968\)  =  \( -1 \cdot 2^{3} \cdot 7 \cdot 47 \cdot 3299 \)
\( I_{10} \)  =  \(-820264960\)  =  \( -1 \cdot 2^{14} \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)
\( J_2 \)  =  \(133\)  =  \( 7 \cdot 19 \)
\( J_4 \)  =  \(-8354\)  =  \( -1 \cdot 2 \cdot 4177 \)
\( J_6 \)  =  \(356384\)  =  \( 2^{5} \cdot 7 \cdot 37 \cdot 43 \)
\( J_8 \)  =  \(-5597561\)  =  \( -1 \cdot 5597561 \)
\( J_{10} \)  =  \(-200260\)  =  \( -1 \cdot 2^{2} \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)
\( g_1 \)  =  \(-2190305047/10540\)
\( g_2 \)  =  \(517208671/5270\)
\( g_3 \)  =  \(-82948376/2635\)
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,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0],C![7,-839,3],C![7,602,3]];

Known rational points: (-1 : -2 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : 1 : 0), (7 : -839 : 3), (7 : 602 : 3)

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank*: \(2\)

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

2-Selmer rank: \(3\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 2 (p = 2), 1 (p = 5), 1 (p = 17), 1 (p = 19), 1 (p = 31)

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

Torsion: \(\Z/{2}\Z\)

2-torsion field: splitting field of \(x^{6} - 20 x^{4} - 26 x^{3} - 255 x^{2} - 1730 x - 1706\) with Galois group $S_4\times C_2$

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 \(\Q\).