Properties

Label 800.a
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 800.a

Label Equation
800.a.1600.1 \(y^2 + (x^3 + x^2 + x + 1)y = -x^4 - x^2\)
800.a.8000.1 \(y^2 + (x^3 + x^2 + x + 1)y = -x^6 + 2x^4 + 4x^3 + 2x^2 - 1\)
800.a.409600.1 \(y^2 = x^6 - 2x^2 + 1\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(5\)\( ( 1 - T )( 1 + T )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( ( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)
\(7\)\( ( 1 - 2 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)\( ( 1 - 4 T + 11 T^{2} )( 1 + 11 T^{2} )\)
\(13\)\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )\)
\(17\)\( ( 1 - 2 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )\)
\(19\)\( ( 1 - 4 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )\)
\(23\)\( ( 1 - 6 T + 23 T^{2} )( 1 - 4 T + 23 T^{2} )\)
\(29\)\( ( 1 - 6 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )\)
$\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.