Genus 2 isogeny class 40000.e

• Label: 40000.e
• Conductor: $40000$
• Sato-Tate group: $J(C_4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\C)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\mathsf{CM})\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Genus 2 curves in isogeny class 40000.e

Label	Equation
40000.e.200000.1	\(y^2 + x^3y = x^5 - 5x^3 - 10x^2 - 8x - 2\)

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 + 2 T + 2 T^{2}\)
\(5\)	\( 1\)

Good L-factors:

Prime	L-Factor
\(3\)	\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}\)
\(7\)	\( 1 + 49 T^{4}\)
\(11\)	\( ( 1 + 6 T + 11 T^{2} )^{2}\)
\(13\)	\( 1 + 169 T^{4}\)
\(17\)	\( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4}\)
\(19\)	\( 1 - 34 T^{2} + 361 T^{4}\)
\(23\)	\( 1 + 529 T^{4}\)
\(29\)	\( ( 1 + 29 T^{2} )^{2}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(C_4)$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 4.4.8000.1 with defining polynomial:
  \(x^{4} - 10 x^{2} + 20\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 4.4.8000.1-25.1-b

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 \(K = \Q (a) \simeq \) 8.0.64000000.1 with defining polynomial \(x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16\)

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.