Properties

Label 1637.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 1637.a

Label Equation
1637.a.1637.1 \(y^2 + y = x^5 - x^4 + x^3 - x^2\)

L-function data

Analytic rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(1637\)\( ( 1 - T )( 1 + 44 T + 1637 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4}\)
\(3\)\( 1 + 2 T + 4 T^{2} + 6 T^{3} + 9 T^{4}\)
\(5\)\( ( 1 + 5 T^{2} )( 1 + 4 T + 5 T^{2} )\)
\(7\)\( 1 + 12 T^{2} + 49 T^{4}\)
\(11\)\( 1 - T + T^{2} - 11 T^{3} + 121 T^{4}\)
\(13\)\( 1 + T + 7 T^{2} + 13 T^{3} + 169 T^{4}\)
\(17\)\( 1 + T + 17 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 10 T^{2} + 361 T^{4}\)
\(23\)\( 1 + 3 T + 23 T^{2} + 69 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 3 T + 43 T^{2} - 87 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

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.