Properties

Label 5113.a
Conductor $5113$
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

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Genus 2 curves in isogeny class 5113.a

Label Equation
5113.a.5113.1 \(y^2 + (x^2 + x + 1)y = x^6 - 2x^4 - x\)

L-function data

Analytic rank:\(2\)  (upper bound)
Mordell-Weil rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(5113\)\( ( 1 + T )( 1 - 33 T + 5113 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}\)
\(3\)\( ( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )\)
\(5\)\( ( 1 + T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )\)
\(7\)\( 1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 4 T + 15 T^{2} + 44 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)\( 1 + 6 T^{2} + 289 T^{4}\)
\(19\)\( 1 + 8 T + 36 T^{2} + 152 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 2 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )\)
\(29\)\( 1 + T - 8 T^{2} + 29 T^{3} + 841 T^{4}\)
$\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.