Properties

 Label 13689.c Conductor $13689$ 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 13689.c

Label Equation
13689.c.41067.1 $$y^2 + (x^3 + x)y = x^5 - 6x^3 - 13x^2 - 6x + 9$$

L-function data

Analytic rank:$$1$$
Mordell-Weil rank:$$1$$

Bad L-factors:
Prime L-Factor
$$3$$$$1 + T$$
$$13$$$$1 + 3 T + 13 T^{2}$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{3} + 4 T^{4}$$
$$5$$$$( 1 + 5 T^{2} )( 1 + 3 T + 5 T^{2} )$$
$$7$$$$( 1 - 3 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )$$
$$11$$$$1 - 2 T + 10 T^{2} - 22 T^{3} + 121 T^{4}$$
$$17$$$$1 - T + 18 T^{2} - 17 T^{3} + 289 T^{4}$$
$$19$$$$1 + T - 12 T^{2} + 19 T^{3} + 361 T^{4}$$
$$23$$$$1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4}$$
$$29$$$$( 1 + 29 T^{2} )( 1 + 9 T + 29 T^{2} )$$
$\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.