Genus 2 curve 10000.b.800000.1

• 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

Minimal equation

$y^2 = x^5 + 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(10000\)	\(=\)	\( 2^{4} \cdot 5^{4} \)
Discriminant:	\( \Delta \)	\(=\)	\(800000\)	\(=\)	\( 2^{8} \cdot 5^{5} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(0\)	\(=\)	\( 0 \)
\( I_4 \)	\(=\)	\(0\)	\(=\)	\( 0 \)
\( I_6 \)	\(=\)	\(0\)	\(=\)	\( 0 \)
\( I_{10} \)	\(=\)	\(1\)	\(=\)	\( 1 \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_{10}$

Rational points

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

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{10}\Z\)

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

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 all of $\GSp(4,\F_\ell)$.

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