Genus 2 isogeny class 8192.b

• Label: 8192.b
• Conductor: $8192$
• Sato-Tate group: $J(E_2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Genus 2 curves in isogeny class 8192.b

Label	Equation
8192.b.131072.1	\(y^2 = x^5 - 3x^4 + 6x^2 - 4x\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

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

Good L-factors:

Prime	L-Factor
\(3\)	\( 1 - 2 T^{2} + 9 T^{4}\)
\(5\)	\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )\)
\(7\)	\( ( 1 - 4 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)	\( 1 - 18 T^{2} + 121 T^{4}\)
\(13\)	\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )\)
\(17\)	\( ( 1 - 2 T + 17 T^{2} )^{2}\)
\(19\)	\( 1 - 34 T^{2} + 361 T^{4}\)
\(23\)	\( ( 1 - 4 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )\)
\(29\)	\( ( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_2)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

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

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 2.0.4.1-512.1-b
  Elliptic curve isogeny class 2.0.4.1-512.1-a

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial \(x^{4} + 1\)

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.