Genus 2 curve 6081.b.164187.1

• Label: 6081.b.164187.1
• Conductor: $6081$
• Discriminant: $164187$
• Mordell-Weil group: \(\Z \oplus \Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(6081\)	\(=\)	\( 3 \cdot 2027 \)
Discriminant:	\( \Delta \)	\(=\)	\(164187\)	\(=\)	\( 3^{4} \cdot 2027 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(804\)	\(=\)	\( 2^{2} \cdot 3 \cdot 67 \)
\( I_4 \)	\(=\)	\(62697\)	\(=\)	\( 3 \cdot 20899 \)
\( I_6 \)	\(=\)	\(11560485\)	\(=\)	\( 3 \cdot 5 \cdot 797 \cdot 967 \)
\( I_{10} \)	\(=\)	\(21015936\)	\(=\)	\( 2^{7} \cdot 3^{4} \cdot 2027 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((0 : 1 : 1)\)	\((1 : 0 : 1)\)	\((-1 : -1 : 1)\)
\((-2 : 0 : 1)\)	\((-1 : 0 : 2)\)	\((0 : -2 : 1)\)	\((1 : -3 : 1)\)	\((-2 : 3 : 1)\)	\((-4 : 5 : 1)\)
\((-1 : -9 : 2)\)	\((-4 : 42 : 1)\)	\((-13 : 112 : 6)\)	\((-13 : 855 : 6)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.080053\)	\(\infty\)
\((-2 : 3 : 1) + (-1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x + z) (x + 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-3xz^2 - 3z^3\)	\(0.044233\)	\(\infty\)

2-torsion field: 6.2.129728.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.003504 \)
Real period:	\( 20.37907 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.285653 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(1\)	\(4\)	\(4\)	\(( 1 - T )( 1 + 3 T + 3 T^{2} )\)
\(2027\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 52 T + 2027 T^{2} )\)

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 \(\Q\).