Genus 2 isogeny class 4096.b

• Label: 4096.b
• Conductor: $4096$
• Sato-Tate group: $J(C_2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\C)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\mathsf{CM})\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 - 2 T^{2} + 9 T^{4}\)	2.3.a_ac
\(5\)	\( ( 1 - 5 T^{2} )^{2}\)	2.5.a_ak
\(7\)	\( ( 1 + 7 T^{2} )^{2}\)	2.7.a_o
\(11\)	\( 1 + 14 T^{2} + 121 T^{4}\)	2.11.a_o
\(13\)	\( ( 1 - 13 T^{2} )^{2}\)	2.13.a_aba
\(17\)	\( ( 1 + 6 T + 17 T^{2} )^{2}\)	2.17.m_cs
\(19\)	\( 1 - 34 T^{2} + 361 T^{4}\)	2.19.a_abi
\(23\)	\( ( 1 + 23 T^{2} )^{2}\)	2.23.a_bu
\(29\)	\( ( 1 - 29 T^{2} )^{2}\)	2.29.a_acg
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(C_2)$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\)

Decomposition of the Jacobian

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

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 2.2.8.1-64.1-a

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

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q(\sqrt{-2}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\C)\)

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 4096.b

Label	Equation
4096.b.65536.1	\(y^2 = x^5 - x\)