Properties

Label 65520.b
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 65520.b

Label Equation
65520.b.131040.1 \(y^2 + (x^3 + x)y = -10x^6 - 82x^4 - 227x^2 - 210\)

L-function data

Analytic rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + T\)
\(3\)\( ( 1 - T )( 1 + T )\)
\(5\)\( ( 1 + T )( 1 - 2 T + 5 T^{2} )\)
\(7\)\( ( 1 + T )( 1 + 7 T^{2} )\)
\(13\)\( ( 1 - T )( 1 + 2 T + 13 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(11\)\( ( 1 + 11 T^{2} )( 1 + 4 T + 11 T^{2} )\)
\(17\)\( ( 1 - 2 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )\)
\(19\)\( ( 1 + 19 T^{2} )( 1 + 4 T + 19 T^{2} )\)
\(23\)\( ( 1 + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\)
\(29\)\( ( 1 - 10 T + 29 T^{2} )( 1 - 6 T + 29 T^{2} )\)
$\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.