Properties

Label 1253.a.1253.1
Conductor 1253
Discriminant -1253
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![-330, 15, 43, -33, 0, 2, -1], R![1, 0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-330, 15, 43, -33, 0, 2, -1]), R([1, 0, 1, 1]))
 

$y^2 + (x^3 + x^2 + 1)y = -x^6 + 2x^5 - 33x^3 + 43x^2 + 15x - 330$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 1253 \)  =  \( 7 \cdot 179 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-1253\)  =  \( -1 \cdot 7 \cdot 179 \)

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);
 

\( I_2 \)  =  \(-827064\)  =  \( -1 \cdot 2^{3} \cdot 3^{3} \cdot 7 \cdot 547 \)
\( I_4 \)  =  \(37524148644\)  =  \( 2^{2} \cdot 3 \cdot 3127012387 \)
\( I_6 \)  =  \(-7994893805035992\)  =  \( -1 \cdot 2^{3} \cdot 3^{3} \cdot 739 \cdot 24019 \cdot 2085257 \)
\( I_{10} \)  =  \(-5132288\)  =  \( -1 \cdot 2^{12} \cdot 7 \cdot 179 \)
\( J_2 \)  =  \(-103383\)  =  \( -1 \cdot 3^{3} \cdot 7 \cdot 547 \)
\( J_4 \)  =  \(54458647\)  =  \( 54458647 \)
\( J_6 \)  =  \(97243994481\)  =  \( 3 \cdot 29 \cdot 79 \cdot 179 \cdot 79043 \)
\( J_8 \)  =  \(-3254780028624958\)  =  \( -1 \cdot 2 \cdot 47 \cdot 34625319453457 \)
\( J_{10} \)  =  \(-1253\)  =  \( -1 \cdot 7 \cdot 179 \)
\( g_1 \)  =  \(1687126365978608485162449/179\)
\( g_2 \)  =  \(8596391751971448839127/179\)
\( g_3 \)  =  \(-829487756384515053\)
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 except over $\R$.

magma: [];
 

There are no rational points.

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*: twice a square

Regulator: 1.0

Real period: 0.20746393501932595902618920284

Tamagawa numbers: 1 (p = 7), 1 (p = 179)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.0.80192.1

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 \(\Q\).