Properties

Label 529.a
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 529.a

Label Equation
529.a.529.1 \(y^2 + (x^3 + x + 1)y = -x^5\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(23\)\( ( 1 - T )^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}\)
\(3\)\( 1 + T^{2} + 9 T^{4}\)
\(5\)\( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}\)
\(7\)\( 1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 6 T + 26 T^{2} + 66 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 3 T + 13 T^{2} )^{2}\)
\(17\)\( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 + 2 T + 19 T^{2} )^{2}\)
\(29\)\( ( 1 + 3 T + 29 T^{2} )^{2}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $G_{3,3}$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)\)

Endomorphisms of the Jacobian

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

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

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