Properties

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

Related objects

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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

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

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.