Genus 2 curve 83422.a.166844.1

• Label: 83422.a.166844.1
• Conductor: $83422$
• Discriminant: $166844$
• 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

$y^2 + (x^3 + 1)y = -x^5 + x^4 + x^2 - 7x + 6$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(83422\)	\(=\)	\( 2 \cdot 53 \cdot 787 \)
Discriminant:	\( \Delta \)	\(=\)	\(166844\)	\(=\)	\( 2^{2} \cdot 53 \cdot 787 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(876\)	\(=\)	\( 2^{2} \cdot 3 \cdot 73 \)
\( I_4 \)	\(=\)	\(12345\)	\(=\)	\( 3 \cdot 5 \cdot 823 \)
\( I_6 \)	\(=\)	\(4168563\)	\(=\)	\( 3 \cdot 7 \cdot 198503 \)
\( I_{10} \)	\(=\)	\(-21356032\)	\(=\)	\( - 2^{9} \cdot 53 \cdot 787 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((1 : 0 : 1)\)	\((0 : 2 : 1)\)	\((1 : -2 : 1)\)	\((0 : -3 : 1)\)
\((-1 : -4 : 1)\)	\((-1 : 4 : 1)\)	\((2 : -4 : 1)\)	\((2 : -5 : 1)\)	\((-3 : -10 : 1)\)	\((3 : -17 : 2)\)
\((3 : -18 : 2)\)	\((-3 : 36 : 1)\)	\((7 : -142 : 3)\)	\((8 : -185 : 3)\)	\((7 : -228 : 3)\)	\((8 : -354 : 3)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((1 : 0 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 + z^3\)	\(0.363641\)	\(\infty\)
\((2 : -5 : 1) - (1 : 0 : 0)\)	\(z (x - 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 + 3z^3\)	\(0.323642\)	\(\infty\)
\((1 : -2 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.235621\)	\(\infty\)

2-torsion field: 6.2.667376.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.025288 \)
Real period:	\( 16.09825 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.814204 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + T + 2 T^{2} )\)
\(53\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + T + 53 T^{2} )\)
\(787\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 43 T + 787 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .

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 \(\Q\).