Properties

Label 6081.b
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 6081.b

Label Equation
6081.b.164187.1 \(y^2 + (x^3 + x^2 + 1)y = -2x^4 - 5x^3 + 5x + 2\)

L-function data

Analytic rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(3\)\( ( 1 - T )( 1 + 3 T + 3 T^{2} )\)
\(2027\)\( ( 1 - T )( 1 - 52 T + 2027 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}\)
\(5\)\( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4}\)
\(7\)\( 1 + 4 T + 12 T^{2} + 28 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 4 T + 6 T^{2} + 52 T^{3} + 169 T^{4}\)
\(17\)\( 1 + 2 T + 27 T^{2} + 34 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 5 T + 22 T^{2} + 95 T^{3} + 361 T^{4}\)
\(23\)\( 1 + T - 3 T^{2} + 23 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 45 T^{2} + 841 T^{4}\)
$\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.