Genus 2 curve 23120.a.92480.1

• Label: 23120.a.92480.1
• Conductor: $23120$
• Discriminant: $92480$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + (x^3 + x)y = -4x^4 + 15x^2 - 20$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(23120\)	\(=\)	\( 2^{4} \cdot 5 \cdot 17^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(92480\)	\(=\)	\( 2^{6} \cdot 5 \cdot 17^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(8216\)	\(=\)	\( 2^{3} \cdot 13 \cdot 79 \)
\( I_4 \)	\(=\)	\(17752\)	\(=\)	\( 2^{3} \cdot 7 \cdot 317 \)
\( I_6 \)	\(=\)	\(48220124\)	\(=\)	\( 2^{2} \cdot 12055031 \)
\( I_{10} \)	\(=\)	\(369920\)	\(=\)	\( 2^{8} \cdot 5 \cdot 17^{2} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2^2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_2^2$

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-2 : 4 : 1)\)	\((2 : -4 : 1)\)	\((-2 : 6 : 1)\)	\((2 : -6 : 1)\)
\((-3 : 11 : 1)\)	\((3 : -11 : 1)\)	\((-3 : 19 : 1)\)	\((3 : -19 : 1)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-2 : 4 : 1) - (1 : 0 : 0)\)	\(z (x + 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - 4z^3\)	\(0.463015\)	\(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.410269\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 - 5z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-3xz^2\)	\(0\)	\(2\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + xz - 4z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-3xz^2 + 2z^3\)	\(0\)	\(2\)

2-torsion field: \(\Q(\sqrt{5}, \sqrt{17})\)

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(4\)
Regulator:	\( 0.189961 \)
Real period:	\( 15.70751 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 0.745954 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(4\)	\(6\)	\(4\)	\(1 + T\)
\(5\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(17\)	\(2\)	\(2\)	\(1\)	\(( 1 - T )^{2}\)

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?
\(2\)	2.180.7	yes
\(3\)	3.90.1	no

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 170.b
  Elliptic curve isogeny class 136.a

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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