Properties

Label 1197.a
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Genus 2 curves in isogeny class 1197.a

Label Equation
1197.a.10773.1 \(y^2 + (x^3 + x^2)y = -x^3 - x^2 - x + 2\)
1197.a.410571.1 \(y^2 + (x^2 + 1)y = x^5 + 12x^4 - 7x^3 - 3x^2 + x\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(3\)\( 1 + T^{2}\)
\(7\)\( ( 1 + T )( 1 - 3 T + 7 T^{2} )\)
\(19\)\( ( 1 + T )( 1 + 19 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 - T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(5\)\( 1 - T - 5 T^{3} + 25 T^{4}\)
\(11\)\( 1 + 5 T + 18 T^{2} + 55 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)\( ( 1 - 6 T + 17 T^{2} )( 1 + 7 T + 17 T^{2} )\)
\(23\)\( ( 1 - 4 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\)
\(29\)\( ( 1 - 10 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )\)
$\cdots$$\cdots$
 
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Sato-Tate group

\(\mathrm{ST} =\) $\mathrm{USp}(4)$

Endomorphisms of the Jacobian

Not 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.