Properties

Label 1532.a.392192.1
Conductor 1532
Discriminant 392192
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, 12, -53, 0, 7, 1], R![1, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 12, -53, 0, 7, 1]), R([1, 1, 1]))
 

$y^2 + (x^2 + x + 1)y = x^5 + 7x^4 - 53x^2 + 12x - 1$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 1532 \)  =  \( 2^{2} \cdot 383 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(392192\)  =  \( 2^{10} \cdot 383 \)

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);
 

\( I_2 \)  =  \(105000\)  =  \( 2^{3} \cdot 3 \cdot 5^{4} \cdot 7 \)
\( I_4 \)  =  \(133343172\)  =  \( 2^{2} \cdot 3^{4} \cdot 17 \cdot 43 \cdot 563 \)
\( I_6 \)  =  \(4137931714248\)  =  \( 2^{3} \cdot 3^{2} \cdot 7529 \cdot 7633321 \)
\( I_{10} \)  =  \(1606418432\)  =  \( 2^{22} \cdot 383 \)
\( J_2 \)  =  \(13125\)  =  \( 3 \cdot 5^{4} \cdot 7 \)
\( J_4 \)  =  \(5788743\)  =  \( 3 \cdot 1929581 \)
\( J_6 \)  =  \(3113886477\)  =  \( 3 \cdot 13 \cdot 79843243 \)
\( J_8 \)  =  \(1840053622644\)  =  \( 2^{2} \cdot 3^{3} \cdot 17037533543 \)
\( J_{10} \)  =  \(392192\)  =  \( 2^{10} \cdot 383 \)
\( g_1 \)  =  \(389490222930908203125/392192\)
\( g_2 \)  =  \(13088268780029296875/392192\)
\( g_3 \)  =  \(536415600139453125/392192\)
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![1,0,0]];
 

All rational points: (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

Regulator: 1.0

Real period: 0.51845243072121604298406669004

Tamagawa numbers: 1 (p = 2), 1 (p = 383)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 5.1.24512.1

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 \(\Q\).