Properties

Label 1109.c
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

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Genus 2 curves in isogeny class 1109.c

Label Equation
1109.c.1109.1 \(y^2 + (x^3 + x)y = x^5 - 2x^3 - 2x^2 - 1\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(1109\)\( ( 1 + T )( 1 - 60 T + 1109 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 - T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(3\)\( ( 1 - 2 T + 3 T^{2} )( 1 + T + 3 T^{2} )\)
\(5\)\( 1 + T + 3 T^{2} + 5 T^{3} + 25 T^{4}\)
\(7\)\( ( 1 - 3 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )\)
\(11\)\( 1 - 5 T + 13 T^{2} - 55 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 7 T + 27 T^{2} + 91 T^{3} + 169 T^{4}\)
\(17\)\( 1 + 2 T - 6 T^{2} + 34 T^{3} + 289 T^{4}\)
\(19\)\( 1 - 2 T + 18 T^{2} - 38 T^{3} + 361 T^{4}\)
\(23\)\( 1 - 6 T + 34 T^{2} - 138 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 3 T - 7 T^{2} - 87 T^{3} + 841 T^{4}\)
$\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.