Genus 2 curve 23289.a.489069.1

• Label: 23289.a.489069.1
• Conductor: $23289$
• Discriminant: $489069$
• 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

$y^2 + y = 3x^5 - 7x^4 + 6x^3 - 2x^2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(23289\)	\(=\)	\( 3 \cdot 7 \cdot 1109 \)
Discriminant:	\( \Delta \)	\(=\)	\(489069\)	\(=\)	\( 3^{2} \cdot 7^{2} \cdot 1109 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(32\)	\(=\)	\( 2^{5} \)
\( I_4 \)	\(=\)	\(1984\)	\(=\)	\( 2^{6} \cdot 31 \)
\( I_6 \)	\(=\)	\(59968\)	\(=\)	\( 2^{6} \cdot 937 \)
\( I_{10} \)	\(=\)	\(-1956276\)	\(=\)	\( - 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 1109 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)	\((1 : -1 : 1)\)	\((2 : -3 : 3)\)
\((3 : 17 : 1)\)	\((3 : -18 : 1)\)	\((2 : -24 : 3)\)	\((-1 : -30 : 4)\)	\((-1 : -34 : 4)\)	\((5 : 75 : 1)\)
\((5 : -76 : 1)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((1 : 0 : 1) + (2 : -3 : 3) - 2 \cdot(1 : 0 : 0)\)	\((x - z) (3x - 2z)\)	\(=\)	\(0,\)	\(3y\)	\(=\)	\(xz^2 - z^3\)	\(0.140331\)	\(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(x^2 - xz + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2 - z^3\)	\(0.087533\)	\(\infty\)

2-torsion field: 5.1.17744.3

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.012241 \)
Real period:	\( 11.70305 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.573028 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(7\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + T + 7 T^{2} )\)
\(1109\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 4 T + 1109 T^{2} )\)

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?
\(2\)	2.6.1	no

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 \(\Q\).