# Properties

 Label 10075.b Conductor $10075$ 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 10075.b

Label Equation
10075.b.654875.1 $$y^2 + xy = x^5 + 2x^4 - 4x^3 - 8x^2 - 1$$

## L-function data

Analytic rank:$$0$$
Mordell-Weil rank:$$0$$

Bad L-factors:
Prime L-Factor
$$5$$$$1 + 2 T + 5 T^{2}$$
$$13$$$$( 1 + T )( 1 - 6 T + 13 T^{2} )$$
$$31$$$$( 1 - T )( 1 + 4 T + 31 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{3} + 4 T^{4}$$
$$3$$$$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$$7$$$$( 1 - 4 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )$$
$$11$$$$( 1 - 4 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$
$$17$$$$1 - T + 16 T^{2} - 17 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - 5 T + 19 T^{2} )( 1 + 19 T^{2} )$$
$$23$$$$( 1 - 8 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )$$
$$29$$$$( 1 + 2 T + 29 T^{2} )^{2}$$
$\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.