Genus 2 curve 12544.i.614656.1

• Label: 12544.i.614656.1
• Conductor: $12544$
• Discriminant: $614656$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $E_3$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 = x^5 + 4x^4 - 13x^3 + 9x^2 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(12544\)	\(=\)	\( 2^{8} \cdot 7^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(614656\)	\(=\)	\( 2^{8} \cdot 7^{4} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(398\)	\(=\)	\( 2 \cdot 199 \)
\( I_4 \)	\(=\)	\(9016\)	\(=\)	\( 2^{3} \cdot 7^{2} \cdot 23 \)
\( I_6 \)	\(=\)	\(912086\)	\(=\)	\( 2 \cdot 7^{2} \cdot 41 \cdot 227 \)
\( I_{10} \)	\(=\)	\(2401\)	\(=\)	\( 7^{4} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_6$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$

Rational points

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

Number of rational Weierstrass points: \(3\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((1 : 0 : 1) - (1 : 0 : 0)\)	\(x - z\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: \(\Q(\zeta_{7})^+\)

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 19.01761 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 1.188600 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	Root number*	L-factor	Cluster picture	Tame reduction?
\(2\)	\(8\)	\(8\)	\(1\)	\(1^*\)	\(1\)		no
\(7\)	\(2\)	\(4\)	\(1\)	\(1\)	\(1 - 4 T + 7 T^{2}\)		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?
\(2\)	2.240.1	yes
\(3\)	3.2880.16	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$E_3$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{7})^+\) with defining polynomial:
  \(x^{3} - x^{2} - 2 x + 1\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -15349824 b^{2} + 8518608 b + 34491744\)
  \(g_6 = 17221522752 b^{2} - 9557224992 b - 38696416416\)
   Conductor norm: 4096

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{7})^+\) with defining polynomial \(x^{3} - x^{2} - 2 x + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)