Properties

Label 5599.a.5599.1
Conductor 5599
Discriminant 5599
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, -2, 1, 0, -1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 1, 0, -1]), R([1, 1, 0, 1]))
 

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 5599 \)  =  \( 11 \cdot 509 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(5599\)  =  \( 11 \cdot 509 \)

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);
 

\( I_2 \)  =  \(-56\)  =  \( -1 \cdot 2^{3} \cdot 7 \)
\( I_4 \)  =  \(3748\)  =  \( 2^{2} \cdot 937 \)
\( I_6 \)  =  \(-848856\)  =  \( -1 \cdot 2^{3} \cdot 3 \cdot 113 \cdot 313 \)
\( I_{10} \)  =  \(22933504\)  =  \( 2^{12} \cdot 11 \cdot 509 \)
\( J_2 \)  =  \(-7\)  =  \( -1 \cdot 7 \)
\( J_4 \)  =  \(-37\)  =  \( -1 \cdot 37 \)
\( J_6 \)  =  \(1397\)  =  \( 11 \cdot 127 \)
\( J_8 \)  =  \(-2787\)  =  \( -1 \cdot 3 \cdot 929 \)
\( J_{10} \)  =  \(5599\)  =  \( 11 \cdot 509 \)
\( g_1 \)  =  \(-16807/5599\)
\( g_2 \)  =  \(12691/5599\)
\( g_3 \)  =  \(6223/509\)
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![-2,1,1],C![-2,8,1],C![-1,-1,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![5,-261,6],C![5,-260,6]];
 

Known rational points: (-2 : 1 : 1), (-2 : 8 : 1), (-1 : -1 : 1), (-1 : 2 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -2 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (5 : -261 : 6), (5 : -260 : 6)

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: 0.0121627345019

Real period: 21.606456118821925109863340696

Tamagawa numbers: 1 (p = 11), 1 (p = 509)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.2.358336.1

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 \(\Q\).