Properties

Label 14724.a.88344.1
Conductor 14724
Discriminant 88344
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, -1, -1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 1, 0, -1, -1]), R([1, 1, 0, 1]))
 

$y^2 + (x^3 + x + 1)y = -x^5 - x^4 + x^2 - x$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 14724 \)  =  \( 2^{2} \cdot 3^{2} \cdot 409 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(88344\)  =  \( 2^{3} \cdot 3^{3} \cdot 409 \)

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);
 

\( I_2 \)  =  \(264\)  =  \( 2^{3} \cdot 3 \cdot 11 \)
\( I_4 \)  =  \(30276\)  =  \( 2^{2} \cdot 3^{2} \cdot 29^{2} \)
\( I_6 \)  =  \(273096\)  =  \( 2^{3} \cdot 3^{2} \cdot 3793 \)
\( I_{10} \)  =  \(361857024\)  =  \( 2^{15} \cdot 3^{3} \cdot 409 \)
\( J_2 \)  =  \(33\)  =  \( 3 \cdot 11 \)
\( J_4 \)  =  \(-270\)  =  \( -1 \cdot 2 \cdot 3^{3} \cdot 5 \)
\( J_6 \)  =  \(2500\)  =  \( 2^{2} \cdot 5^{4} \)
\( J_8 \)  =  \(2400\)  =  \( 2^{5} \cdot 3 \cdot 5^{2} \)
\( J_{10} \)  =  \(88344\)  =  \( 2^{3} \cdot 3^{3} \cdot 409 \)
\( g_1 \)  =  \(1449459/3272\)
\( g_2 \)  =  \(-179685/1636\)
\( g_3 \)  =  \(75625/2454\)
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![-6,-129,5],C![-6,370,5],C![-2,-2,1],C![-2,11,1],C![-1,-1,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-11,2],C![1,-2,1],C![1,-2,2],C![1,-1,0],C![1,-1,1],C![1,0,0]];
 

Known rational points: (-6 : -129 : 5), (-6 : 370 : 5), (-2 : -2 : 1), (-2 : 11 : 1), (-1 : -1 : 1), (-1 : 2 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -11 : 2), (1 : -2 : 1), (1 : -2 : 2), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0)

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: \(2\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.00632089500916

Real period: 16.321468793249390209436891989

Tamagawa numbers: 3 (p = 2), 2 (p = 3), 1 (p = 409)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.2.628224.1

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 \(\Q\).