Properties

Label 3125.a.3125.1
Conductor 3125
Discriminant 3125
Mordell-Weil group \(\Z/{5}\Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^5$ (homogenize, simplify)
$y^2 + z^3y = x^5z$ (dehomogenize, simplify)
$y^2 = 4x^5 + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, 0, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, 0, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, 0, 0, 0, 0, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(0\) =  \( 0 \)
\( I_4 \)  = \(0\) =  \( 0 \)
\( I_6 \)  = \(0\) =  \( 0 \)
\( I_{10} \)  = \(12800000\) =  \( 2^{12} \cdot 5^{5} \)
\( J_2 \)  = \(0\) =  \( 0 \)
\( J_4 \)  = \(0\) =  \( 0 \)
\( J_6 \)  = \(0\) =  \( 0 \)
\( J_8 \)  = \(0\) =  \( 0 \)
\( J_{10} \)  = \(3125\) =  \( 5^{5} \)
\( g_1 \)  = \(0\)
\( g_2 \)  = \(0\)
\( g_3 \)  = \(0\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_{10}$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

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

Number of rational Weierstrass points: \(1\)

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

This curve is locally solvable everywhere.

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);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{5}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(5\)

2-torsion field: 5.1.50000.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 17.95784 \)
Tamagawa product: \( 1 \)
Torsion order:\( 5 \)
Leading coefficient: \( 0.718313 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5\) \(5\) \(5\) \(1\) \(1\)

Sato-Tate group

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

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

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