Genus 2 curve 336.a.172032.1

• Label: 336.a.172032.1
• Conductor: $336$
• Discriminant: $-172032$
• Mordell-Weil group: \(\Z/{2}\Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + (x^3 + x)y = -x^6 + 15x^4 - 75x^2 - 56$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(336\)	\(=\)	\( 2^{4} \cdot 3 \cdot 7 \)
Discriminant:	\( \Delta \)	\(=\)	\(-172032\)	\(=\)	\( - 2^{13} \cdot 3 \cdot 7 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(16916\)	\(=\)	\( 2^{2} \cdot 4229 \)
\( I_4 \)	\(=\)	\(151117825\)	\(=\)	\( 5^{2} \cdot 6044713 \)
\( I_6 \)	\(=\)	\(232872423961\)	\(=\)	\( 397 \cdot 586580413 \)
\( I_{10} \)	\(=\)	\(-21504\)	\(=\)	\( - 2^{10} \cdot 3 \cdot 7 \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2^2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_2^2$

Rational points

This curve has no rational points.

Number of rational Weierstrass points: \(0\)

This curve is locally solvable except over $\Q_{2}$.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - D_\infty\)	\(3x^2 - 32z^2\)	\(=\)	\(0,\)	\(6y\)	\(=\)	\(-35xz^2\)	\(0\)	\(2\)

2-torsion field: 8.4.260112384.2

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 0.356066 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 0.178033 \)
Analytic order of Ш:	\( 2 \)   (rounded)
Order of Ш:	twice a square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	Root number*	L-factor	Cluster picture	Tame reduction?
\(2\)	\(4\)	\(13\)	\(1\)	\(-1^*\)	\(1 + T\)		no
\(3\)	\(1\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 2 T + 3 T^{2} )\)		yes
\(7\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(( 1 - T )( 1 + 7 T^{2} )\)		yes

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.45.1	yes
\(3\)	3.720.5	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 14.a
  Elliptic curve isogeny class 24.a

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

Additional information

The Jacobian of this curve is conjectured to have the smallest conductor among all genus 2 curves whose analytic order of Sha is not a square.