Properties

Label 1300.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 1300.a

Label Equation
1300.a.130000.1 \(y^2 + (x^3 + x)y = 2x^4 + 9x^2 + 13\)

L-function data

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