Properties

Label 289238.a.289238.1
Conductor 289238
Discriminant -289238
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![10, 40, 51, 20, 0, 1, -1], R![1, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([10, 40, 51, 20, 0, 1, -1]), R([1, 1, 1]))

$y^2 + (x^2 + x + 1)y = -x^6 + x^5 + 20x^3 + 51x^2 + 40x + 10$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 289238 \)  =  \( 2 \cdot 17 \cdot 47 \cdot 181 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-289238\)  =  \( -1 \cdot 2 \cdot 17 \cdot 47 \cdot 181 \)

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);

\( I_2 \)  =  \(102312\)  =  \( 2^{3} \cdot 3^{2} \cdot 7^{2} \cdot 29 \)
\( I_4 \)  =  \(-375420\)  =  \( -1 \cdot 2^{2} \cdot 3 \cdot 5 \cdot 6257 \)
\( I_6 \)  =  \(-12763734264\)  =  \( -1 \cdot 2^{3} \cdot 3^{2} \cdot 23 \cdot 7707569 \)
\( I_{10} \)  =  \(-1184718848\)  =  \( -1 \cdot 2^{13} \cdot 17 \cdot 47 \cdot 181 \)
\( J_2 \)  =  \(12789\)  =  \( 3^{2} \cdot 7^{2} \cdot 29 \)
\( J_4 \)  =  \(6818849\)  =  \( 6818849 \)
\( J_6 \)  =  \(4850280481\)  =  \( 179 \cdot 431 \cdot 62869 \)
\( J_8 \)  =  \(3883383846677\)  =  \( 31 \cdot 78941 \cdot 1586887 \)
\( J_{10} \)  =  \(-289238\)  =  \( -1 \cdot 2 \cdot 17 \cdot 47 \cdot 181 \)
\( g_1 \)  =  \(-342123524046146462949/289238\)
\( g_2 \)  =  \(-14263326884806825581/289238\)
\( g_3 \)  =  \(-793304701907528601/289238\)
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: [];

No rational points known.

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(1\)

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

2-Selmer rank: \(3\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 1 (p = 2), 1 (p = 17), 1 (p = 47), 1 (p = 181)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: splitting field of \(x^{6} - 6 x^{4} - 6 x^{3} - 2 x^{2} + 8 x + 11\) with Galois group $S_6$

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 \(\Q\).