# Properties

 Label 10125.a Conductor $10125$ 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

# Learn more

## Genus 2 curves in isogeny class 10125.a

Label Equation
10125.a.10125.1 $$y^2 + (x^3 + x + 1)y = -x^2 + x - 1$$
10125.a.10125.2 $$y^2 + (x^3 + x^2 + 1)y = 9x^4 + 28x^3 - x^2 - 34x + 13$$

## L-function data

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

Bad L-factors:
Prime L-Factor
$$3$$$$1 + T + 3 T^{2}$$
$$5$$$$1 + T$$

Good L-factors:
Prime L-Factor
$$2$$$$1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}$$
$$7$$$$( 1 - 3 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$$11$$$$1 + 3 T + 17 T^{2} + 33 T^{3} + 121 T^{4}$$
$$13$$$$1 + 2 T + 17 T^{2} + 26 T^{3} + 169 T^{4}$$
$$17$$$$( 1 - 8 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )$$
$$19$$$$1 - 2 T^{2} + 361 T^{4}$$
$$23$$$$1 - 3 T + 12 T^{2} - 69 T^{3} + 529 T^{4}$$
$$29$$$$1 + 5 T + 33 T^{2} + 145 T^{3} + 841 T^{4}$$
$\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.