Genus 2 isogeny class 12544.g

• Label: 12544.g
• Conductor: $12544$
• Sato-Tate group: $J(E_4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)
\(7\)	\( 1 + 7 T^{2}\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}\)	2.3.e_i
\(5\)	\( 1 + 8 T^{2} + 25 T^{4}\)	2.5.a_i
\(11\)	\( 1 - 6 T^{2} + 121 T^{4}\)	2.11.a_ag
\(13\)	\( 1 + 8 T^{2} + 169 T^{4}\)	2.13.a_i
\(17\)	\( 1 + 2 T + 2 T^{2} + 34 T^{3} + 289 T^{4}\)	2.17.c_c
\(19\)	\( 1 + 4 T + 8 T^{2} + 76 T^{3} + 361 T^{4}\)	2.19.e_i
\(23\)	\( ( 1 - 4 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )\)	2.23.a_be
\(29\)	\( ( 1 - 8 T + 29 T^{2} )( 1 + 8 T + 29 T^{2} )\)	2.29.a_ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_4)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 4.0.2744.1 with defining polynomial:
  \(x^{4} - x^{3} + 3 x^{2} + 3 x + 2\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -56 b^{3} + 112 b^{2} + 280 b + 112\)
  \(g_6 = -812 b^{3} + 8596 b^{2} + 7700 b + 5124\)
   Conductor norm: 1024
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -280 b^{3} + 560 b^{2} - 1288 b + 112\)
  \(g_6 = -16338 b^{3} + 25704 b^{2} - 59150 b - 22764\)
   Conductor norm: 1024

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 8.0.481890304.3 with defining polynomial \(x^{8} - 2 x^{7} - 3 x^{6} + 8 x^{5} + x^{4} - 2 x^{3} + 23 x^{2} - 12 x + 2\)

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.

Genus 2 curves in isogeny class 12544.g

Label	Equation
12544.g.175616.1	\(y^2 + x^3y = x^5 + x^4 - 2x^2 - 4x - 2\)