Genus 2 isogeny class 12544.a

• Label: 12544.a
• Conductor: $12544$
• Sato-Tate group: $E_6$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Genus 2 curves in isogeny class 12544.a

Label	Equation
12544.a.12544.1	\(y^2 = x^5 + 2x^4 - x^3 - 3x^2 - x\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)
\(7\)	\( 1 - 4 T + 7 T^{2}\)

Good L-factors:

Prime	L-Factor
\(3\)	\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)	\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4}\)
\(11\)	\( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4}\)
\(13\)	\( ( 1 - 13 T^{2} )^{2}\)
\(17\)	\( 1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4}\)
\(19\)	\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)	\( 1 - 15 T + 98 T^{2} - 345 T^{3} + 529 T^{4}\)
\(29\)	\( ( 1 + 6 T + 29 T^{2} )^{2}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_6$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{28})^+\) with defining polynomial:
  \(x^{6} - 7 x^{4} + 14 x^{2} - 7\)

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
  \(g_4 = 880 b^{5} + 959 b^{4} - 4900 b^{3} - 4851 b^{2} + 6734 b + 6209\)
  \(g_6 = -41104 b^{5} - \frac{46697}{2} b^{4} + 290997 b^{3} + \frac{425957}{2} b^{2} - 465297 b - 380856\)
   Conductor norm: 4096

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{28})^+\) with defining polynomial \(x^{6} - 7 x^{4} + 14 x^{2} - 7\)

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.