Properties

Label 59411.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

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Genus 2 curves in isogeny class 59411.a

Label Equation
59411.a.59411.1 \(y^2 + y = x^6 - x^5 - x^3 + 2x^2 - x\)

L-function data

Analytic rank:\(3\)
 
Bad L-factors:
Prime L-Factor
\(11\)\( 1 + 6 T + 11 T^{2}\)
\(491\)\( ( 1 + T )( 1 - 22 T + 491 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(3\)\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 3 T + 7 T^{2} + 15 T^{3} + 25 T^{4}\)
\(7\)\( 1 + 6 T + 20 T^{2} + 42 T^{3} + 49 T^{4}\)
\(13\)\( 1 + 7 T + 31 T^{2} + 91 T^{3} + 169 T^{4}\)
\(17\)\( 1 - T + 11 T^{2} - 17 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 + 4 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )\)
\(23\)\( 1 - 3 T + 22 T^{2} - 69 T^{3} + 529 T^{4}\)
\(29\)\( ( 1 - 3 T + 29 T^{2} )( 1 + 7 T + 29 T^{2} )\)
$\cdots$$\cdots$
 
See L-function page for more information

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.