Properties

Label 12500.a.12500.1
Conductor 12500
Discriminant 12500
Mordell-Weil group \(\Z/{5}\Z\)
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

Related objects

Learn more about

Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = x^6 + 2x^3 - x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^6 + 2x^3z^3 - xz^5$ (dehomogenize, simplify)
$y^2 = 5x^6 + 10x^3 - 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 2, 0, 0, 1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 2, 0, 0, 1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -4, 0, 10, 0, 0, 5]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

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

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

Rational points

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

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

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
\((-1 : 0 : 1) + (0 : 0 : 1) - D_\infty\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(5\)

2-torsion field: 5.1.200000.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 25.01138 \)
Tamagawa product: \( 1 \)
Torsion order:\( 5 \)
Leading coefficient: \( 1.000455 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(2\) \(2\) \(1\) \(1 + T^{2}\)
\(5\) \(5\) \(5\) \(1\) \(1\)

Sato-Tate group

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

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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{5}) \) with defining polynomial \(x^{2} - x - 1\)

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

Endomorphism ring over \(\overline{\Q}\):

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