Properties

Label 1038.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 1038.a

Label Equation
1038.a.1038.2 \(y^2 + (x^3 + 1)y = x^4 + 2x^2 + x + 1\)
1038.a.1038.1 \(y^2 + (x^2 + x)y = x^5 - 12x^4 + 26x^3 + 46x^2 + 21x + 3\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( ( 1 + T )( 1 - T + 2 T^{2} )\)
\(3\)\( ( 1 + T )( 1 - T + 3 T^{2} )\)
\(173\)\( ( 1 + T )( 1 - 6 T + 173 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}\)
\(7\)\( 1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4}\)
\(11\)\( ( 1 + 11 T^{2} )^{2}\)
\(13\)\( ( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)\( 1 - 4 T + 10 T^{2} - 68 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 - 4 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )\)
\(23\)\( 1 - T - 2 T^{2} - 23 T^{3} + 529 T^{4}\)
\(29\)\( ( 1 - 6 T + 29 T^{2} )( 1 + 8 T + 29 T^{2} )\)
$\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.