Properties

Label 46234.a.92468.1
Conductor 46234
Discriminant 92468
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

Learn more about

Show commands for: Magma / SageMath

Minimal equation

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

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 46234 \)  =  \( 2 \cdot 23117 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(92468\)  =  \( 2^{2} \cdot 23117 \)

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);
 

\( I_2 \)  =  \(552\)  =  \( 2^{3} \cdot 3 \cdot 23 \)
\( I_4 \)  =  \(77028\)  =  \( 2^{2} \cdot 3 \cdot 7^{2} \cdot 131 \)
\( I_6 \)  =  \(6317160\)  =  \( 2^{3} \cdot 3 \cdot 5 \cdot 61 \cdot 863 \)
\( I_{10} \)  =  \(378748928\)  =  \( 2^{14} \cdot 23117 \)
\( J_2 \)  =  \(69\)  =  \( 3 \cdot 23 \)
\( J_4 \)  =  \(-604\)  =  \( -1 \cdot 2^{2} \cdot 151 \)
\( J_6 \)  =  \(5172\)  =  \( 2^{2} \cdot 3 \cdot 431 \)
\( J_8 \)  =  \(-1987\)  =  \( -1 \cdot 1987 \)
\( J_{10} \)  =  \(92468\)  =  \( 2^{2} \cdot 23117 \)
\( g_1 \)  =  \(1564031349/92468\)
\( g_2 \)  =  \(-49604859/23117\)
\( g_3 \)  =  \(6155973/23117\)
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![-3,2,1],C![-3,24,1],C![-2,0,1],C![-2,7,1],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-117,6],C![1,-100,6],C![1,-5,2],C![1,-4,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-8,1],C![2,-1,1],C![7,-1575,12],C![7,-496,12]];
 

Known rational points: (-3 : 2 : 1), (-3 : 24 : 1), (-2 : 0 : 1), (-2 : 7 : 1), (-1 : -2 : 1), (-1 : 2 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -117 : 6), (1 : -100 : 6), (1 : -5 : 2), (1 : -4 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (2 : -8 : 1), (2 : -1 : 1), (7 : -1575 : 12), (7 : -496 : 12)

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank*: \(3\)

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

2-Selmer rank: \(3\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.0158961772285

Real period: 19.586114739900558979373124189

Tamagawa numbers: 2 (p = 2), 1 (p = 23117)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.2.1479488.1

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 \(\Q\).