Genus 2 curve 3564.b.705672.1

• Label: 3564.b.705672.1
• Conductor: $3564$
• Discriminant: $-705672$
• Mordell-Weil group: \(\Z \oplus \Z/{3}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(3564\)	\(=\)	\( 2^{2} \cdot 3^{4} \cdot 11 \)
Discriminant:	\( \Delta \)	\(=\)	\(-705672\)	\(=\)	\( - 2^{3} \cdot 3^{6} \cdot 11^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(404\)	\(=\)	\( 2^{2} \cdot 101 \)
\( I_4 \)	\(=\)	\(6537\)	\(=\)	\( 3 \cdot 2179 \)
\( I_6 \)	\(=\)	\(761861\)	\(=\)	\( 761861 \)
\( I_{10} \)	\(=\)	\(371712\)	\(=\)	\( 2^{10} \cdot 3 \cdot 11^{2} \)

Automorphism group

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

Rational points

All points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((0 : 1 : 1)\)	\((-1 : 1 : 1)\)	\((0 : -2 : 1)\)
\((3 : 1 : 1)\)	\((-3 : -1 : 2)\)	\((3 : -32 : 1)\)	\((-3 : 32 : 2)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{3}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 - 2xz + 3z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(4z^3\)	\(0.017792\)	\(\infty\)
\(2 \cdot(0 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2 + z^3\)	\(0\)	\(3\)

2-torsion field: 6.0.41472.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(1\)
Regulator:	\( 0.017792 \)
Real period:	\( 12.92260 \)
Tamagawa product:	\( 18 \)
Torsion order:	\( 3 \)
Leading coefficient:	\( 0.459848 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(2\)	\(3\)	\(3\)	\(1 + T + T^{2}\)
\(3\)	\(4\)	\(6\)	\(3\)	\(1 + T + 3 T^{2}\)
\(11\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 3 T + 11 T^{2} )\)

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?
\(2\)	2.40.3	no
\(3\)	3.80.1	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 \(\Q\).