Genus 2 curve 1740.a.104400.1

• Label: 1740.a.104400.1
• Conductor: $1740$
• Discriminant: $-104400$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{6}\Z\)
• 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

$y^2 + (x^2 + x)y = 2x^5 - 14x^3 - 5x^2 + 30x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(1740\)	\(=\)	\( 2^{2} \cdot 3 \cdot 5 \cdot 29 \)
Discriminant:	\( \Delta \)	\(=\)	\(-104400\)	\(=\)	\( - 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 29 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(28100\)	\(=\)	\( 2^{2} \cdot 5^{2} \cdot 281 \)
\( I_4 \)	\(=\)	\(7231657\)	\(=\)	\( 7231657 \)
\( I_6 \)	\(=\)	\(99549877317\)	\(=\)	\( 3 \cdot 11 \cdot 1657 \cdot 1820557 \)
\( I_{10} \)	\(=\)	\(-13363200\)	\(=\)	\( - 2^{11} \cdot 3^{2} \cdot 5^{2} \cdot 29 \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (2 : -2 : 1),\, (2 : -4 : 1),\, (15 : -1380 : 8)\)

Number of rational Weierstrass points: \(3\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{6}\Z\)

Generator	$D_0$						Height	Order
\((0 : 0 : 1) + (15 : -1380 : 8) - 2 \cdot(1 : 0 : 0)\)	\(x (8x - 15z)\)	\(=\)	\(0,\)	\(16y\)	\(=\)	\(-23xz^2\)	\(0\)	\(2\)
\((2 : -4 : 1) + (15 : -1380 : 8) - 2 \cdot(1 : 0 : 0)\)	\((x - 2z) (8x - 15z)\)	\(=\)	\(0,\)	\(16y\)	\(=\)	\(-167xz^2 + 270z^3\)	\(0\)	\(6\)

2-torsion field: 3.1.116.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 5.116046 \)
Tamagawa product:	\( 16 \)
Torsion order:	\( 12 \)
Leading coefficient:	\( 0.568449 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(2\)	\(4\)	\(4\)	\(( 1 - T )( 1 + T )\)
\(3\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 - T + 3 T^{2} )\)
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 3 T + 5 T^{2} )\)
\(29\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 6 T + 29 T^{2} )\)

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
\(3\)	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 \(\Q\).