Properties

Label 975.a.63375.1
Conductor 975
Discriminant -63375
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, 1, 2, 1, 0, -1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, 2, 1, 0, -1]), R([1, 0, 0, 1]))
 

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 975 \)  =  \( 3 \cdot 5^{2} \cdot 13 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-63375\)  =  \( -1 \cdot 3 \cdot 5^{3} \cdot 13^{2} \)

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);
 

\( I_2 \)  =  \(296\)  =  \( 2^{3} \cdot 37 \)
\( I_4 \)  =  \(-194300\)  =  \( -1 \cdot 2^{2} \cdot 5^{2} \cdot 29 \cdot 67 \)
\( I_6 \)  =  \(-32609400\)  =  \( -1 \cdot 2^{3} \cdot 3 \cdot 5^{2} \cdot 17 \cdot 23 \cdot 139 \)
\( I_{10} \)  =  \(-259584000\)  =  \( -1 \cdot 2^{12} \cdot 3 \cdot 5^{3} \cdot 13^{2} \)
\( J_2 \)  =  \(37\)  =  \( 37 \)
\( J_4 \)  =  \(2081\)  =  \( 2081 \)
\( J_6 \)  =  \(35929\)  =  \( 19 \cdot 31 \cdot 61 \)
\( J_8 \)  =  \(-750297\)  =  \( -1 \cdot 3 \cdot 383 \cdot 653 \)
\( J_{10} \)  =  \(-63375\)  =  \( -1 \cdot 3 \cdot 5^{3} \cdot 13^{2} \)
\( g_1 \)  =  \(-69343957/63375\)
\( g_2 \)  =  \(-105408893/63375\)
\( g_3 \)  =  \(-49186801/63375\)
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![-1,0,1],C![1,-1,0],C![1,0,0],C![3,-14,1]];
 

All rational points: (-1 : 0 : 1), (1 : -1 : 0), (1 : 0 : 0), (3 : -14 : 1)

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

Number of rational Weierstrass points: \(2\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 14.356290407763699883663167478

Tamagawa numbers: 1 (p = 3), 2 (p = 5), 2 (p = 13)

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

Torsion: \(\Z/{2}\Z \times \Z/{6}\Z\)

2-torsion field: \(\Q(\sqrt{-3}, \sqrt{5})\)

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 \(\Q\).