# Properties

 Label 10040.c Conductor $10040$ 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 10040.c

Label Equation
10040.c.502000.1 $$y^2 + (x^2 + 1)y = x^5 - 3x^4 + 3x^3 - 2x$$

## L-function data

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

Bad L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{2}$$
$$5$$$$( 1 - T )( 1 + 4 T + 5 T^{2} )$$
$$251$$$$( 1 - T )( 1 - 6 T + 251 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 4 T^{2} + 9 T^{4}$$
$$7$$$$1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4}$$
$$11$$$$1 - 2 T - 4 T^{2} - 22 T^{3} + 121 T^{4}$$
$$13$$$$1 + 6 T + 24 T^{2} + 78 T^{3} + 169 T^{4}$$
$$17$$$$1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4}$$
$$19$$$$1 + 16 T^{2} + 361 T^{4}$$
$$23$$$$1 + 6 T + 18 T^{2} + 138 T^{3} + 529 T^{4}$$
$$29$$$$1 + 6 T + 28 T^{2} + 174 T^{3} + 841 T^{4}$$
$\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.