Properties

Label 886.a
Conductor $886$
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Genus 2 curves in isogeny class 886.a

Label Equation
886.a.3544.1 \(y^2 + (x^3 + x)y = -x^4 - x + 1\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( ( 1 - T )( 1 + 2 T + 2 T^{2} )\)
\(443\)\( ( 1 - T )( 1 - 24 T + 443 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + T + T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)\( 1 - T - 5 T^{2} - 5 T^{3} + 25 T^{4}\)
\(7\)\( 1 + 5 T + 15 T^{2} + 35 T^{3} + 49 T^{4}\)
\(11\)\( ( 1 - 2 T + 11 T^{2} )( 1 + 3 T + 11 T^{2} )\)
\(13\)\( 1 - 2 T + 8 T^{2} - 26 T^{3} + 169 T^{4}\)
\(17\)\( 1 + 9 T + 43 T^{2} + 153 T^{3} + 289 T^{4}\)
\(19\)\( 1 - 4 T + 18 T^{2} - 76 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 6 T + 23 T^{2} )( 1 + 6 T + 23 T^{2} )\)
\(29\)\( ( 1 - 10 T + 29 T^{2} )( 1 + 3 T + 29 T^{2} )\)
$\cdots$$\cdots$
 
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Sato-Tate group

\(\mathrm{ST} =\) $\mathrm{USp}(4)$

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

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.