Properties

Label 1147.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 1147.a

Label Equation
1147.a.35557.1 \(y^2 + xy = x^5 + 8x^4 + 18x^3 + 8x^2 + x\)
1147.a.35557.2 \(y^2 + xy = x^5 + 6x^4 - 32x^2 + x\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(31\)\( ( 1 + T )( 1 + 31 T^{2} )\)
\(37\)\( ( 1 + T )( 1 - 6 T + 37 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + T + 2 T^{3} + 4 T^{4}\)
\(3\)\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )\)
\(7\)\( 1 - 3 T + 6 T^{2} - 21 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 3 T + 2 T^{2} + 33 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )\)
\(17\)\( ( 1 - 2 T + 17 T^{2} )( 1 + 4 T + 17 T^{2} )\)
\(19\)\( 1 - 2 T - 2 T^{2} - 38 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\)
\(29\)\( 1 - 2 T - 14 T^{2} - 58 T^{3} + 841 T^{4}\)
$\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.