Properties

Label 12500.a.12500.1
Conductor 12500
Discriminant 12500
Sato-Tate group $N(G_{3,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Show commands for: Magma / SageMath

Minimal equation

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

$y^2 + (x^3 + 1)y = x^6 + 2x^3 - x$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 12500 \)  =  \( 2^{2} \cdot 5^{5} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(12500\)  =  \( 2^{2} \cdot 5^{5} \)

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);

\( I_2 \)  =  \(-600\)  =  \( -1 \cdot 2^{3} \cdot 3 \cdot 5^{2} \)
\( I_4 \)  =  \(202500\)  =  \( 2^{2} \cdot 3^{4} \cdot 5^{4} \)
\( I_6 \)  =  \(-67995000\)  =  \( -1 \cdot 2^{3} \cdot 3^{2} \cdot 5^{4} \cdot 1511 \)
\( I_{10} \)  =  \(51200000\)  =  \( 2^{14} \cdot 5^{5} \)
\( J_2 \)  =  \(-75\)  =  \( -1 \cdot 3 \cdot 5^{2} \)
\( J_4 \)  =  \(-1875\)  =  \( -1 \cdot 3 \cdot 5^{4} \)
\( J_6 \)  =  \(73125\)  =  \( 3^{2} \cdot 5^{4} \cdot 13 \)
\( J_8 \)  =  \(-2250000\)  =  \( -1 \cdot 2^{4} \cdot 3^{2} \cdot 5^{6} \)
\( J_{10} \)  =  \(12500\)  =  \( 2^{2} \cdot 5^{5} \)
\( g_1 \)  =  \(-759375/4\)
\( g_2 \)  =  \(253125/4\)
\( g_3 \)  =  \(131625/4\)
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![0,-1,1],C![0,0,1]];

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

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

Number of rational Weierstrass points: \(1\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(0\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

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

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

Torsion: \(\Z/{5}\Z\)

2-torsion field: 5.1.200000.1

Sato-Tate group

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

Decomposition

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

Endomorphisms

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{5}) \) with defining polynomial \(x^{2} - x - 1\)

Of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{5}}{2}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \R\)