Genus 2 curve 50000.b.800000.1

• Label: 50000.b.800000.1
• Conductor: $50000$
• Discriminant: $800000$
• Mordell-Weil group: \(\Z \oplus \Z/{5}\Z\)
• Sato-Tate group: $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{RM}\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 + y = x^6 - 4x^5 + 5x^3 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(50000\)	\(=\)	\( 2^{4} \cdot 5^{5} \)
Discriminant:	\( \Delta \)	\(=\)	\(800000\)	\(=\)	\( 2^{8} \cdot 5^{5} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(100\)	\(=\)	\( 2^{2} \cdot 5^{2} \)
\( I_4 \)	\(=\)	\(625\)	\(=\)	\( 5^{4} \)
\( I_6 \)	\(=\)	\(14905\)	\(=\)	\( 5 \cdot 11 \cdot 271 \)
\( I_{10} \)	\(=\)	\(32\)	\(=\)	\( 2^{5} \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3 - 2x^2z\)	\(0.566737\)	\(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3\)	\(0\)	\(5\)

2-torsion field: 5.1.800000.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(1\)
Regulator:	\( 0.566737 \)
Real period:	\( 17.11758 \)
Tamagawa product:	\( 5 \)
Torsion order:	\( 5 \)
Leading coefficient:	\( 1.940235 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

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

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?
\(2\)	2.6.2	no
\(3\)	3.216.1	no

Sato-Tate group

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

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 \(\overline{\Q}\):

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