Properties

Label 3391.b.3391.1
Conductor 3391
Discriminant -3391
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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Minimal equation

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

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 3391 \)  =  \( 3391 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-3391\)  =  \( -1 \cdot 3391 \)

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);
 

\( I_2 \)  =  \(-504\)  =  \( -1 \cdot 2^{3} \cdot 3^{2} \cdot 7 \)
\( I_4 \)  =  \(420\)  =  \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
\( I_6 \)  =  \(-845784\)  =  \( -1 \cdot 2^{3} \cdot 3^{2} \cdot 17 \cdot 691 \)
\( I_{10} \)  =  \(-13889536\)  =  \( -1 \cdot 2^{12} \cdot 3391 \)
\( J_2 \)  =  \(-63\)  =  \( -1 \cdot 3^{2} \cdot 7 \)
\( J_4 \)  =  \(161\)  =  \( 7 \cdot 23 \)
\( J_6 \)  =  \(813\)  =  \( 3 \cdot 271 \)
\( J_8 \)  =  \(-19285\)  =  \( -1 \cdot 5 \cdot 7 \cdot 19 \cdot 29 \)
\( J_{10} \)  =  \(-3391\)  =  \( -1 \cdot 3391 \)
\( g_1 \)  =  \(992436543/3391\)
\( g_2 \)  =  \(40257567/3391\)
\( g_3 \)  =  \(-3226797/3391\)
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![-1,-1,2],C![-1,0,1],C![-1,1,1],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![2,-48,3],C![2,-5,3]];
 

Known rational points: (-1 : -2 : 2), (-1 : -1 : 2), (-1 : 0 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -2 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (2 : -48 : 3), (2 : -5 : 3)

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: 0.00847272373257

Real period: 22.961203925404113032827165628

Tamagawa numbers: 1 (p = 3391)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.4.217024.1

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 \(\Q\).