Genus 2 curve 2156.b.34496.1

• Label: 2156.b.34496.1
• Conductor: $2156$
• Discriminant: $34496$
• Mordell-Weil group: \(\Z \oplus \Z/{6}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(2156\)	\(=\)	\( 2^{2} \cdot 7^{2} \cdot 11 \)
Discriminant:	\( \Delta \)	\(=\)	\(34496\)	\(=\)	\( 2^{6} \cdot 7^{2} \cdot 11 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(2916\)	\(=\)	\( 2^{2} \cdot 3^{6} \)
\( I_4 \)	\(=\)	\(41745\)	\(=\)	\( 3 \cdot 5 \cdot 11^{2} \cdot 23 \)
\( I_6 \)	\(=\)	\(37024569\)	\(=\)	\( 3^{2} \cdot 563 \cdot 7307 \)
\( I_{10} \)	\(=\)	\(4415488\)	\(=\)	\( 2^{13} \cdot 7^{2} \cdot 11 \)

Automorphism group

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

Rational points

All points
\((1 : 0 : 0)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((0 : 1 : 1)\)	\((1 : 0 : 1)\)	\((-1 : 0 : 2)\)
\((1 : -2 : 1)\)	\((-1 : 2 : 2)\)

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{6}\Z\)

Generator	$D_0$						Height	Order
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.123038\)	\(\infty\)
\((-1 : 2 : 2) + (1 : -2 : 1) - 2 \cdot(1 : 0 : 0)\)	\((x - z) (2x + z)\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-3xz^2 - z^3\)	\(0\)	\(6\)

2-torsion field: 4.0.2156.2

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.123038 \)
Real period:	\( 21.05583 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 6 \)
Leading coefficient:	\( 0.287853 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(2\)	\(6\)	\(4\)	\(( 1 + T )^{2}\)
\(7\)	\(2\)	\(2\)	\(1\)	\(( 1 - T )( 1 + T )\)
\(11\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 11 T^{2} )\)

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?
\(2\)	2.90.3	yes
\(3\)	3.720.4	yes

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 154.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 \(\Q\).