Genus 2 isogeny class 20736.k

• Label: 20736.k
• Conductor: $20736$
• 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

Genus 2 curves in isogeny class 20736.k

Label	Equation
20736.k.373248.1	\(y^2 + x^3y = 2\)

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)
\(3\)	\( 1\)

Good L-factors:

Prime	L-Factor
\(5\)	\( ( 1 + 5 T^{2} )^{2}\)
\(7\)	\( ( 1 + 4 T + 7 T^{2} )^{2}\)
\(11\)	\( ( 1 + 11 T^{2} )^{2}\)
\(13\)	\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )\)
\(17\)	\( ( 1 + 17 T^{2} )^{2}\)
\(19\)	\( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)	\( ( 1 + 23 T^{2} )^{2}\)
\(29\)	\( ( 1 + 29 T^{2} )^{2}\)
$\cdots$	$\cdots$

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 576.e
  Elliptic curve isogeny class 36.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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}, \sqrt{-3})\) with defining polynomial \(x^{4} + 2 x^{2} + 4\)

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q(\sqrt{-3}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\C)\)

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.