Properties

Label 4489.b
Conductor $4489$
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 4489.b

Label Equation
4489.b.4489.1 \(y^2 + (x^3 + x + 1)y = -x^6 + x^5 - x^4 - x\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(67\)\( 1 + 15 T + 67 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + T + T^{2} + 2 T^{3} + 4 T^{4}\)
\(3\)\( 1 + T + 3 T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}\)
\(7\)\( 1 + 3 T + 13 T^{2} + 21 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 10 T^{2} + 121 T^{4}\)
\(13\)\( 1 + T + 9 T^{2} + 13 T^{3} + 169 T^{4}\)
\(17\)\( 1 + T + 23 T^{2} + 17 T^{3} + 289 T^{4}\)
\(19\)\( 1 - 3 T + 39 T^{2} - 57 T^{3} + 361 T^{4}\)
\(23\)\( 1 - 2 T + 18 T^{2} - 46 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 5 T + 47 T^{2} - 145 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

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.