Properties

Label 1285.a
Conductor $1285$
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 1285.a

Label Equation
1285.a.1285.1 \(y^2 + y = x^5 - 2x^4 + 3x^3 - x\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(5\)\( ( 1 + T )( 1 + T + 5 T^{2} )\)
\(257\)\( ( 1 + T )( 1 + 8 T + 257 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T^{2} + 4 T^{4}\)
\(3\)\( 1 - T + T^{2} - 3 T^{3} + 9 T^{4}\)
\(7\)\( 1 + T - 2 T^{2} + 7 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 3 T + 17 T^{2} + 33 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 6 T + 13 T^{2} )( 1 + 13 T^{2} )\)
\(17\)\( ( 1 - 4 T + 17 T^{2} )( 1 + 7 T + 17 T^{2} )\)
\(19\)\( 1 - 6 T + 24 T^{2} - 114 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 3 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )\)
\(29\)\( 1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}\)
$\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.