Properties

Label 10000.b.800000.1
Conductor $10000$
Discriminant $800000$
Mordell-Weil group \(\Z/{10}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

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

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: \( N \)  \(=\)  \(10000\) \(=\) \( 2^{4} \cdot 5^{4} \)
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: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1)\)
All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1)\)
All points: \((1 : 0 : 0),\, (0 : -1/2 : 1),\, (0 : 1/2 : 1),\, (-1 : 0 : 1)\)

magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,0,0]]; // minimal model
 
magma: [C![-1,0,1],C![0,-1/2,1],C![0,1/2,1],C![1,0,0]]; // 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: \(\Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0\) \(10\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0\) \(10\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 1/2 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(1/2xz^2 + 1/2z^3\) \(0\) \(10\)

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

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 10.31407 \)
Tamagawa product: \( 10 \)
Torsion order:\( 10 \)
Leading coefficient: \( 1.031407 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.180.2 yes
\(3\) 3.1296.1 no

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