Properties

Label 3125.a.3125.1
Conductor 3125
Discriminant 3125
Sato-Tate group $F_{ac}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \C\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{CM}\)
\(\overline{\Q}\)-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, 0, 0, 0, 0, 1], R![1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, 0, 1]), R([1]))

$y^2 + y = x^5$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 3125 \)  =  \( 5^{5} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(3125\)  =  \( 5^{5} \)

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);

\( I_2 \)  =  \(0\)  =  \( 0 \)
\( I_4 \)  =  \(0\)  =  \( 0 \)
\( I_6 \)  =  \(0\)  =  \( 0 \)
\( I_{10} \)  =  \(12800000\)  =  \( 2^{12} \cdot 5^{5} \)
\( J_2 \)  =  \(0\)  =  \( 0 \)
\( J_4 \)  =  \(0\)  =  \( 0 \)
\( J_6 \)  =  \(0\)  =  \( 0 \)
\( J_8 \)  =  \(0\)  =  \( 0 \)
\( J_{10} \)  =  \(3125\)  =  \( 5^{5} \)
\( g_1 \)  =  \(0\)
\( g_2 \)  =  \(0\)
\( g_3 \)  =  \(0\)
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);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(C_{10} \) (GAP id : [10,2])

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![0,-1,1],C![0,0,1],C![1,0,0]];

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

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 = 5)

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

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

2-torsion field: 5.1.50000.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $F_{ac}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{U}(1)\)

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{5})\) with defining polynomial \(x^{4} - x^{3} + x^{2} - x + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)the maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q(\zeta_{5})\) (CM)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\C \times \C\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{5}) \) with generator \(a^{3} - a^{2}\) with minimal polynomial \(x^{2} - x - 1\):
\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{5}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
Sato Tate group: $F_{ab}$
of \(\GL_2\)-type, simple