Properties

Label 51035.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 51035.a

Label Equation
51035.a.255175.1 \(y^2 + (x^3 + x + 1)y = -3x^4 + 5x^2 + 2x\)

L-function data

Analytic rank:\(3\)
 
Bad L-factors:
Prime L-Factor
\(5\)\( ( 1 + T )( 1 + T + 5 T^{2} )\)
\(59\)\( ( 1 + T )( 1 + 14 T + 59 T^{2} )\)
\(173\)\( ( 1 - T )( 1 + 2 T + 173 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}\)
\(3\)\( ( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )\)
\(7\)\( ( 1 + 3 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)
\(11\)\( 1 + 5 T + 17 T^{2} + 55 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 3 T + 10 T^{2} + 39 T^{3} + 169 T^{4}\)
\(17\)\( ( 1 + 17 T^{2} )( 1 + 7 T + 17 T^{2} )\)
\(19\)\( 1 + 2 T + 5 T^{2} + 38 T^{3} + 361 T^{4}\)
\(23\)\( 1 + 8 T + 34 T^{2} + 184 T^{3} + 529 T^{4}\)
\(29\)\( 1 - T + 43 T^{2} - 29 T^{3} + 841 T^{4}\)
$\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.