Properties

Label 830.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 830.a

Label Equation
830.a.6640.1 \(y^2 + (x^3 + 1)y = -x^5 + x^4 - 2x^2 + x + 1\)
830.a.830000.1 \(y^2 + (x^2 + x)y = x^5 - 2x^4 + 16x^3 + 8x^2 + x\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( ( 1 - T )( 1 + T + 2 T^{2} )\)
\(5\)\( ( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(83\)\( ( 1 + T )( 1 - 4 T + 83 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}\)
\(7\)\( 1 + 3 T + 6 T^{2} + 21 T^{3} + 49 T^{4}\)
\(11\)\( 1 + T - 6 T^{2} + 11 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 6 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )\)
\(17\)\( 1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 8 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )\)
\(29\)\( 1 - 3 T + 16 T^{2} - 87 T^{3} + 841 T^{4}\)
$\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.