Genus 2 isogeny class 8192.a

• Label: 8192.a
• Conductor: $8192$
• 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:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 - 2 T^{2} + 9 T^{4}\)	2.3.a_ac
\(5\)	\( ( 1 - 4 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )\)	2.5.ac_c
\(7\)	\( ( 1 - 4 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)	2.7.a_ac
\(11\)	\( 1 + 14 T^{2} + 121 T^{4}\)	2.11.a_o
\(13\)	\( 1 + 6 T + 18 T^{2} + 78 T^{3} + 169 T^{4}\)	2.13.g_s
\(17\)	\( ( 1 - 17 T^{2} )^{2}\)	2.17.a_abi
\(19\)	\( 1 + 30 T^{2} + 361 T^{4}\)	2.19.a_be
\(23\)	\( ( 1 - 4 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )\)	2.23.a_be
\(29\)	\( ( 1 - 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )\)	2.29.g_s
$\cdots$	$\cdots$	$\cdots$

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 \(\Q (b) \simeq \) \(\Q(\sqrt[4]{2})\) with defining polynomial:
  \(x^{4} - 2\)

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
  \(g_4 = 160 b^{2} + 144\)
  \(g_6 = -1792 b^{3} - 3456 b\)
   Conductor norm: 32
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 160 b^{2} + 144\)
  \(g_6 = 1792 b^{3} + 3456 b\)
   Conductor norm: 32

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 \(K = \Q (a) \simeq \) 8.0.16777216.2 with defining polynomial \(x^{8} - 4 x^{6} + 8 x^{4} - 4 x^{2} + 1\)

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 8192.a

Label	Equation
8192.a.32768.1	\(y^2 = x^5 - 3x^3 + 2x\)
8192.a.131072.1	\(y^2 + y = 4x^5 + 15x^4 + 8x^3 - 3x^2 - x\)