Genus 2 curve 168328.a.336656.1

• Label: 168328.a.336656.1
• Conductor: $168328$
• Discriminant: $-336656$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z\)
• 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

$y^2 + (x^3 + x)y = x^5 - x^4 - x^3 + x^2 - 2x + 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(168328\)	\(=\)	\( 2^{3} \cdot 53 \cdot 397 \)
Discriminant:	\( \Delta \)	\(=\)	\(-336656\)	\(=\)	\( - 2^{4} \cdot 53 \cdot 397 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(496\)	\(=\)	\( 2^{4} \cdot 31 \)
\( I_4 \)	\(=\)	\(3172\)	\(=\)	\( 2^{2} \cdot 13 \cdot 61 \)
\( I_6 \)	\(=\)	\(816364\)	\(=\)	\( 2^{2} \cdot 409 \cdot 499 \)
\( I_{10} \)	\(=\)	\(1346624\)	\(=\)	\( 2^{6} \cdot 53 \cdot 397 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : -1 : 1)\)	\((0 : 1 : 1)\)	\((-1 : -1 : 1)\)	\((1 : -1 : 1)\)
\((1 : 1 : 2)\)	\((-1 : 3 : 1)\)	\((3 : -1 : 2)\)	\((1 : -6 : 2)\)	\((3 : -38 : 2)\)	\((-5 : 57 : 3)\)
\((-8 : 83 : 1)\)	\((-5 : 113 : 3)\)	\((-8 : 437 : 1)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + xz - z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2xz^2 + z^3\)	\(0.639768\)	\(\infty\)
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.553094\)	\(\infty\)
\((1 : -1 : 1) - (1 : -1 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.159908\)	\(\infty\)

2-torsion field: 5.3.84164.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.046316 \)
Real period:	\( 15.56270 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.441610 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(3\)	\(4\)	\(2\)	\(1 + T + 2 T^{2}\)
\(53\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 4 T + 53 T^{2} )\)
\(397\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 24 T + 397 T^{2} )\)

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 \(\Q\).