Properties

Label 160000.c.800000.1
Conductor $160000$
Discriminant $800000$
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
Sato-Tate group $F_{ac}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \C\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{CM}\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

Learn more

Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 = x^5 - 1$ (homogenize, simplify)
$y^2 = x^5z - z^6$ (dehomogenize, simplify)
$y^2 = x^5 - 1$ (minimize, homogenize)

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

Invariants

Conductor: \( N \)  \(=\)  \(160000\) \(=\) \( 2^{8} \cdot 5^{4} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(800000\) \(=\) \( 2^{8} \cdot 5^{5} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(0\) \(=\)  \( 0 \)
\( I_4 \)  \(=\) \(0\) \(=\)  \( 0 \)
\( I_6 \)  \(=\) \(0\) \(=\)  \( 0 \)
\( I_{10} \)  \(=\) \(1\) \(=\)  \( 1 \)
\( J_2 \)  \(=\) \(0\) \(=\)  \( 0 \)
\( J_4 \)  \(=\) \(0\) \(=\)  \( 0 \)
\( J_6 \)  \(=\) \(0\) \(=\)  \( 0 \)
\( J_8 \)  \(=\) \(0\) \(=\)  \( 0 \)
\( J_{10} \)  \(=\) \(800000\) \(=\)  \( 2^{8} \cdot 5^{5} \)
\( g_1 \)  \(=\) \(0\)
\( g_2 \)  \(=\) \(0\)
\( g_3 \)  \(=\) \(0\)

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

magma: [C![1,0,0],C![1,0,1]]; // minimal model
 
magma: [C![1,0,0],C![1,0,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 2xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(1.207710\) \(\infty\)
\((1 : 0 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 2xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(1.207710\) \(\infty\)
\((1 : 0 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 2xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(1/2xz^2 - 1/2z^3\) \(1.207710\) \(\infty\)
\((1 : 0 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: \(\Q(\zeta_{5})\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 1.207710 \)
Real period: \( 4.612592 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.785339 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(8\) \(1\) \(1\)
\(5\) \(4\) \(5\) \(2\) \(1\)

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