Properties

Label 6201.a.241839.1
Conductor 6201
Discriminant -241839
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![2, -12, 21, -13, 0, 2], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -12, 21, -13, 0, 2]), R([1, 0, 0, 1]))
 

$y^2 + (x^3 + 1)y = 2x^5 - 13x^3 + 21x^2 - 12x + 2$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 6201 \)  =  \( 3^{2} \cdot 13 \cdot 53 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-241839\)  =  \( -1 \cdot 3^{3} \cdot 13^{2} \cdot 53 \)

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);
 

\( I_2 \)  =  \(-2520\)  =  \( -1 \cdot 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
\( I_4 \)  =  \(3476772\)  =  \( 2^{2} \cdot 3^{2} \cdot 13 \cdot 17 \cdot 19 \cdot 23 \)
\( I_6 \)  =  \(-5109650136\)  =  \( -1 \cdot 2^{3} \cdot 3^{2} \cdot 70967363 \)
\( I_{10} \)  =  \(-990572544\)  =  \( -1 \cdot 2^{12} \cdot 3^{3} \cdot 13^{2} \cdot 53 \)
\( J_2 \)  =  \(-315\)  =  \( -1 \cdot 3^{2} \cdot 5 \cdot 7 \)
\( J_4 \)  =  \(-32082\)  =  \( -1 \cdot 2 \cdot 3 \cdot 5347 \)
\( J_6 \)  =  \(5629636\)  =  \( 2^{2} \cdot 1407409 \)
\( J_8 \)  =  \(-700647516\)  =  \( -1 \cdot 2^{2} \cdot 3^{3} \cdot 89 \cdot 72893 \)
\( J_{10} \)  =  \(-241839\)  =  \( -1 \cdot 3^{3} \cdot 13^{2} \cdot 53 \)
\( g_1 \)  =  \(114865340625/8957\)
\( g_2 \)  =  \(-37138925250/8957\)
\( g_3 \)  =  \(-20688912300/8957\)
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![-9,341,2],C![-9,380,2],C![-6,-2093,7],C![-6,1966,7],C![0,-2,1],C![0,1,1],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,-11,1],C![2,2,1]];
 

Known rational points: (-9 : 341 : 2), (-9 : 380 : 2), (-6 : -2093 : 7), (-6 : 1966 : 7), (0 : -2 : 1), (0 : 1 : 1), (1 : -5 : 2), (1 : -4 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (2 : -11 : 1), (2 : 2 : 1)

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: \(3\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.0180095896456

Real period: 18.300730831014328001726048179

Tamagawa numbers: 2 (p = 3), 2 (p = 13), 1 (p = 53)

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

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

2-torsion field: 6.2.19415808.1

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 \(\Q\).