Properties

Label 3125.a
Sato-Tate group $F_{ac}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \C\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{CM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Genus 2 curves in isogeny class 3125.a

Label Equation
3125.a.3125.1 \(y^2 + y = x^5\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(5\)\( 1\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(3\)\( 1 + 9 T^{4}\)
\(7\)\( 1 + 49 T^{4}\)
\(11\)\( 1 + T - 9 T^{2} + 11 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 169 T^{4}\)
\(17\)\( 1 + 289 T^{4}\)
\(19\)\( ( 1 + 19 T^{2} )^{2}\)
\(23\)\( 1 + 529 T^{4}\)
\(29\)\( ( 1 + 29 T^{2} )^{2}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $F_{ac}$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\times\mathrm{U}(1)\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{5})\) with defining polynomial \(x^{4} - x^{3} + x^{2} - x + 1\)

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q(\zeta_{5})\) (CM)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\C \times \C\)

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