Genus 2 isogeny class 25600.e

• Label: 25600.e
• Conductor: $25600$
• Sato-Tate group: $E_4$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)
\(5\)	\( 1 - 2 T + 5 T^{2}\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}\)	2.3.c_c
\(7\)	\( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4}\)	2.7.ag_s
\(11\)	\( 1 - 18 T^{2} + 121 T^{4}\)	2.11.a_as
\(13\)	\( 1 - 6 T + 18 T^{2} - 78 T^{3} + 169 T^{4}\)	2.13.ag_s
\(17\)	\( 1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}\)	2.17.ac_c
\(19\)	\( ( 1 + 4 T + 19 T^{2} )^{2}\)	2.19.i_cc
\(23\)	\( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4}\)	2.23.ac_c
\(29\)	\( ( 1 - 29 T^{2} )^{2}\)	2.29.a_acg
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{20})^+\) with defining polynomial:
  \(x^{4} - 5 x^{2} + 5\)

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
  \(g_4 = -\frac{32303200}{130321} b^{3} + \frac{61624800}{130321} b^{2} + \frac{44011200}{130321} b - \frac{84012560}{130321}\)
  \(g_6 = -\frac{530918514880}{47045881} b^{3} + \frac{1009835128000}{47045881} b^{2} + \frac{734310067200}{47045881} b - \frac{1396419152000}{47045881}\)
   Conductor norm: 4096

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-1}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{20})^+\) with defining polynomial \(x^{4} - 5 x^{2} + 5\)

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 25600.e

Label	Equation
25600.e.128000.1	\(y^2 = x^5 - x^4 - x^2 - x\)