Properties

Label 62924.a.251696.1
Conductor 62924
Discriminant 251696
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![1, -4, 6, -2, -1], R![0, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -4, 6, -2, -1]), R([0, 0, 0, 1]))
 

$y^2 + x^3y = -x^4 - 2x^3 + 6x^2 - 4x + 1$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 62924 \)  =  \( 2^{2} \cdot 15731 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(251696\)  =  \( 2^{4} \cdot 15731 \)

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);
 

\( I_2 \)  =  \(960\)  =  \( 2^{6} \cdot 3 \cdot 5 \)
\( I_4 \)  =  \(105792\)  =  \( 2^{6} \cdot 3 \cdot 19 \cdot 29 \)
\( I_6 \)  =  \(19951104\)  =  \( 2^{9} \cdot 3 \cdot 31 \cdot 419 \)
\( I_{10} \)  =  \(1030946816\)  =  \( 2^{16} \cdot 15731 \)
\( J_2 \)  =  \(120\)  =  \( 2^{3} \cdot 3 \cdot 5 \)
\( J_4 \)  =  \(-502\)  =  \( -1 \cdot 2 \cdot 251 \)
\( J_6 \)  =  \(6096\)  =  \( 2^{4} \cdot 3 \cdot 127 \)
\( J_8 \)  =  \(119879\)  =  \( 313 \cdot 383 \)
\( J_{10} \)  =  \(251696\)  =  \( 2^{4} \cdot 15731 \)
\( g_1 \)  =  \(1555200000/15731\)
\( g_2 \)  =  \(-54216000/15731\)
\( g_3 \)  =  \(5486400/15731\)
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![-45,496,11],C![-45,90629,11],C![-2,-3,1],C![-2,11,1],C![-1,-3,1],C![-1,4,1],C![0,-1,1],C![0,1,1],C![1,-4,2],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1],C![1,3,2],C![2,-17,3],C![2,-5,1],C![2,-3,1],C![2,9,3],C![3,-23,1],C![3,-4,1]];
 

Known rational points: (-45 : 496 : 11), (-45 : 90629 : 11), (-2 : -3 : 1), (-2 : 11 : 1), (-1 : -3 : 1), (-1 : 4 : 1), (0 : -1 : 1), (0 : 1 : 1), (1 : -4 : 2), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (1 : 0 : 1), (1 : 3 : 2), (2 : -17 : 3), (2 : -5 : 1), (2 : -3 : 1), (2 : 9 : 3), (3 : -23 : 1), (3 : -4 : 1)

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: 0.0136624765493

Real period: 17.731932025966203028745355324

Tamagawa numbers: 3 (p = 2), 1 (p = 15731)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.2.4027136.1

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 \(\Q\).