Genus 2 curve 70450.c.704500.1

• Label: 70450.c.704500.1
• Conductor: $70450$
• Discriminant: $-704500$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z \oplus \Z/{2}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(70450\)	\(=\)	\( 2 \cdot 5^{2} \cdot 1409 \)
Discriminant:	\( \Delta \)	\(=\)	\(-704500\)	\(=\)	\( - 2^{2} \cdot 5^{3} \cdot 1409 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1012\)	\(=\)	\( 2^{2} \cdot 11 \cdot 23 \)
\( I_4 \)	\(=\)	\(58105\)	\(=\)	\( 5 \cdot 11621 \)
\( I_6 \)	\(=\)	\(13468045\)	\(=\)	\( 5 \cdot 2693609 \)
\( I_{10} \)	\(=\)	\(-90176000\)	\(=\)	\( - 2^{9} \cdot 5^{3} \cdot 1409 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)	\((-1 : -2 : 1)\)
\((-1 : 2 : 1)\)	\((1 : -2 : 1)\)	\((2 : -2 : 1)\)	\((-3 : 0 : 2)\)	\((2 : -7 : 1)\)	\((-3 : 8 : 1)\)
\((-1 : 9 : 2)\)	\((2 : -9 : 3)\)	\((-1 : -16 : 2)\)	\((-3 : 18 : 1)\)	\((-3 : 19 : 2)\)	\((2 : -26 : 3)\)
\((10 : -145 : 3)\)	\((10 : -882 : 3)\)	\((29 : -1314 : 5)\)	\((29 : -23200 : 5)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((0 : 0 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.446544\)	\(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.411611\)	\(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2z^3\)	\(0.300988\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + xz - z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2\)	\(0\)	\(2\)

2-torsion field: 6.2.3970562000.2

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(4\)
Regulator:	\( 0.046457 \)
Real period:	\( 16.52129 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 0.767540 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + T + 2 T^{2} )\)
\(5\)	\(2\)	\(3\)	\(2\)	\(1 + 3 T + 5 T^{2}\)
\(1409\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 10 T + 1409 T^{2} )\)

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?
\(2\)	2.15.1	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 \(\Q\).