Genus 2 curve 3675.a.385875.1

• Label: 3675.a.385875.1
• Conductor: $3675$
• Discriminant: $-385875$
• Mordell-Weil group: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(3675\)	\(=\)	\( 3 \cdot 5^{2} \cdot 7^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-385875\)	\(=\)	\( - 3^{2} \cdot 5^{3} \cdot 7^{3} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1536\)	\(=\)	\( 2^{9} \cdot 3 \)
\( I_4 \)	\(=\)	\(-11760\)	\(=\)	\( - 2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \)
\( I_6 \)	\(=\)	\(-6032145\)	\(=\)	\( - 3 \cdot 5 \cdot 7^{2} \cdot 29 \cdot 283 \)
\( I_{10} \)	\(=\)	\(-1543500\)	\(=\)	\( - 2^{2} \cdot 3^{2} \cdot 5^{3} \cdot 7^{3} \)

Automorphism group

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

Rational points

All points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((1 : -1 : 1)\)	\((-2 : 2 : 1)\)	\((2 : 4 : 1)\)	\((-3 : 9 : 1)\)
\((2 : -16 : 1)\)	\((-5 : 34 : 2)\)	\((-5 : 41 : 2)\)

Number of rational Weierstrass points: \(3\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

Generator	$D_0$						Height	Order
\((1 : -1 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3\)	\(0.093653\)	\(\infty\)
\((-2 : 2 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - z) (x + 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2\)	\(0\)	\(2\)
\((-3 : 9 : 1) + (-2 : 2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x + 2z) (x + 3z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-7xz^2 - 12z^3\)	\(0\)	\(2\)

2-torsion field: 3.1.140.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.093653 \)
Real period:	\( 9.686160 \)
Tamagawa product:	\( 8 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 0.453571 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 3 T + 3 T^{2} )\)
\(5\)	\(2\)	\(3\)	\(2\)	\(1 + 2 T + 5 T^{2}\)
\(7\)	\(2\)	\(3\)	\(2\)	\(1 + 4 T + 7 T^{2}\)

Galois representations

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

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.120.3	yes

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 \(\Q\).