Properties

Label 7225.a.36125.1
Conductor 7225
Discriminant -36125
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -3, 0, 2, -1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -3, 0, 2, -1]), R([1, 0, 0, 1]))
 

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 7225 \)  =  \( 5^{2} \cdot 17^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-36125\)  =  \( -1 \cdot 5^{3} \cdot 17^{2} \)

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);
 

\( I_2 \)  =  \(1960\)  =  \( 2^{3} \cdot 5 \cdot 7^{2} \)
\( I_4 \)  =  \(-8156\)  =  \( -1 \cdot 2^{2} \cdot 2039 \)
\( I_6 \)  =  \(-20952536\)  =  \( -1 \cdot 2^{3} \cdot 11 \cdot 457 \cdot 521 \)
\( I_{10} \)  =  \(-147968000\)  =  \( -1 \cdot 2^{12} \cdot 5^{3} \cdot 17^{2} \)
\( J_2 \)  =  \(245\)  =  \( 5 \cdot 7^{2} \)
\( J_4 \)  =  \(2586\)  =  \( 2 \cdot 3 \cdot 431 \)
\( J_6 \)  =  \(64636\)  =  \( 2^{2} \cdot 11 \cdot 13 \cdot 113 \)
\( J_8 \)  =  \(2287106\)  =  \( 2 \cdot 19 \cdot 139 \cdot 433 \)
\( J_{10} \)  =  \(-36125\)  =  \( -1 \cdot 5^{3} \cdot 17^{2} \)
\( g_1 \)  =  \(-7061881225/289\)
\( g_2 \)  =  \(-304240314/289\)
\( g_3 \)  =  \(-155191036/1445\)
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,-6,2],C![-1,-1,1],C![-1,-1,2],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0],C![2,-7,1],C![2,-2,1],C![5,-95,3],C![5,-57,3]];
 

Known rational points: (-1 : -6 : 2), (-1 : -1 : 1), (-1 : -1 : 2), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : 0 : 0), (2 : -7 : 1), (2 : -2 : 1), (5 : -95 : 3), (5 : -57 : 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: \(3\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.0374304482509

Real period: 18.345600350189077256668619141

Tamagawa numbers: 2 (p = 5), 1 (p = 17)

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

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

2-torsion field: 8.0.46240000.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition

Simple over \(\overline{\Q}\)

Endomorphisms

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)\(\Z [\sqrt{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{2}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).