Genus 2 curve 3417.a.686817.1

• Label: 3417.a.686817.1
• Conductor: $3417$
• Discriminant: $686817$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• 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

$y^2 + (x^2 + x)y = 3x^5 - 9x^4 + 6x^3 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(3417\)	\(=\)	\( 3 \cdot 17 \cdot 67 \)
Discriminant:	\( \Delta \)	\(=\)	\(686817\)	\(=\)	\( 3^{2} \cdot 17 \cdot 67^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1348\)	\(=\)	\( 2^{2} \cdot 337 \)
\( I_4 \)	\(=\)	\(46057\)	\(=\)	\( 11 \cdot 53 \cdot 79 \)
\( I_6 \)	\(=\)	\(19365765\)	\(=\)	\( 3 \cdot 5 \cdot 29 \cdot 44519 \)
\( I_{10} \)	\(=\)	\(87912576\)	\(=\)	\( 2^{7} \cdot 3^{2} \cdot 17 \cdot 67^{2} \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -1 : 1),\, (-1 : 3 : 3)\)

Number of rational Weierstrass points: \(4\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(4x^2 - 9xz + 4z^2\)	\(=\)	\(0,\)	\(8y\)	\(=\)	\(-13xz^2 + 4z^3\)	\(0\)	\(2\)
\((-1 : 3 : 3) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\)	\((x - z) (3x + z)\)	\(=\)	\(0,\)	\(6y\)	\(=\)	\(-5xz^2 - z^3\)	\(0\)	\(2\)
\((1 : -1 : 1) - (1 : 0 : 0)\)	\(x - z\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0\)	\(2\)

2-torsion field: \(\Q(\sqrt{17}) \)

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(3\)
Regulator:	\( 1 \)
Real period:	\( 11.58029 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 0.723768 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(17\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 2 T + 17 T^{2} )\)
\(67\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 - 4 T + 67 T^{2} )\)

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?
\(2\)	2.360.2	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 \(\Q\).