Genus 2 isogeny class 65536.a

• Label: 65536.a
• Conductor: $65536$
• Sato-Tate group: $D_{2,1}$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\C)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\mathsf{CM})\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( ( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)	2.3.a_c
\(5\)	\( ( 1 + 5 T^{2} )^{2}\)	2.5.a_k
\(7\)	\( ( 1 + 7 T^{2} )^{2}\)	2.7.a_o
\(11\)	\( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )\)	2.11.a_ao
\(13\)	\( ( 1 + 13 T^{2} )^{2}\)	2.13.a_ba
\(17\)	\( ( 1 + 6 T + 17 T^{2} )^{2}\)	2.17.m_cs
\(19\)	\( ( 1 - 2 T + 19 T^{2} )( 1 + 2 T + 19 T^{2} )\)	2.19.a_bi
\(23\)	\( ( 1 + 23 T^{2} )^{2}\)	2.23.a_bu
\(29\)	\( ( 1 + 29 T^{2} )^{2}\)	2.29.a_cg
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $D_{2,1}$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 256.d
  Elliptic curve isogeny class 256.a

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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 \(\overline{\Q}\):

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

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 65536.a

Label	Equation
65536.a.65536.1	\(y^2 = x^5 + x\)