Properties

Label 464.a.29696.1
Conductor 464
Discriminant -29696
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

Related objects

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -2, -4, 3, 8], R![1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -2, -4, 3, 8]), R([1, 1]))
 

$y^2 + (x + 1)y = 8x^5 + 3x^4 - 4x^3 - 2x^2$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 464 \)  =  \( 2^{4} \cdot 29 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-29696\)  =  \( -1 \cdot 2^{10} \cdot 29 \)

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);
 

\( I_2 \)  =  \(5440\)  =  \( 2^{6} \cdot 5 \cdot 17 \)
\( I_4 \)  =  \(-336320\)  =  \( -1 \cdot 2^{6} \cdot 5 \cdot 1051 \)
\( I_6 \)  =  \(-642023936\)  =  \( -1 \cdot 2^{9} \cdot 1253953 \)
\( I_{10} \)  =  \(-121634816\)  =  \( -1 \cdot 2^{22} \cdot 29 \)
\( J_2 \)  =  \(680\)  =  \( 2^{3} \cdot 5 \cdot 17 \)
\( J_4 \)  =  \(22770\)  =  \( 2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 23 \)
\( J_6 \)  =  \(1180736\)  =  \( 2^{6} \cdot 19 \cdot 971 \)
\( J_8 \)  =  \(71106895\)  =  \( 5 \cdot 223 \cdot 63773 \)
\( J_{10} \)  =  \(-29696\)  =  \( -1 \cdot 2^{10} \cdot 29 \)
\( g_1 \)  =  \(-141985700000/29\)
\( g_2 \)  =  \(-6991813125/29\)
\( g_3 \)  =  \(-533176100/29\)
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,-2,2],C![0,-1,1],C![0,0,1],C![1,-6,2],C![1,0,0]];
 

All rational points: (-1 : -2 : 2), (0 : -1 : 1), (0 : 0 : 1), (1 : -6 : 2), (1 : 0 : 0)

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

Number of rational Weierstrass points: \(3\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 14.421430832107244729188977786

Tamagawa numbers: 4 (p = 2), 1 (p = 29)

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

Torsion: \(\Z/{2}\Z \times \Z/{8}\Z\)

2-torsion field: 3.1.116.1

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 \(\Q\).