Genus 2 curve 100315.a.501575.1

• Label: 100315.a.501575.1
• Conductor: $100315$
• Discriminant: $-501575$
• Mordell-Weil group: \(\Z \oplus \Z\)
• Sato-Tate group: $\mathrm{USp}(4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 + (x^3 + x + 1)y = 2x^5 + 3x^4 + x^3 + 2x^2 - 5x + 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(100315\)	\(=\)	\( 5 \cdot 20063 \)
Discriminant:	\( \Delta \)	\(=\)	\(-501575\)	\(=\)	\( - 5^{2} \cdot 20063 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(4380\)	\(=\)	\( 2^{2} \cdot 3 \cdot 5 \cdot 73 \)
\( I_4 \)	\(=\)	\(252849\)	\(=\)	\( 3 \cdot 89 \cdot 947 \)
\( I_6 \)	\(=\)	\(473905527\)	\(=\)	\( 3 \cdot 157968509 \)
\( I_{10} \)	\(=\)	\(64201600\)	\(=\)	\( 2^{7} \cdot 5^{2} \cdot 20063 \)

Automorphism group

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

Rational points

Known points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : 1 : 1),\, (1 : -4 : 1),\, (1 : -5 : 2),\, (1 : -8 : 2)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z\)

Generator	$D_0$						Height	Order
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(2x^2z\)	\(1.038436\)	\(\infty\)
\((1 : -4 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - 3z^3\)	\(0.050193\)	\(\infty\)

2-torsion field: 6.4.1284032.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.052092 \)
Real period:	\( 14.98942 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.561682 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + T + 5 T^{2} )\)
\(20063\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 88 T + 20063 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$\mathrm{USp}(4)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).