# Properties

 Label 10037.a Conductor $10037$ 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 10037.a

Label Equation
10037.a.10037.1 $$y^2 + y = x^5 + x^2$$

## L-function data

Analytic rank:$$2$$  (upper bound)
Mordell-Weil rank:$$2$$

Bad L-factors:
Prime L-Factor
$$10037$$$$( 1 + T )( 1 + 176 T + 10037 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$
$$5$$$$( 1 + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$7$$$$( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$$13$$$$1 + 5 T + 21 T^{2} + 65 T^{3} + 169 T^{4}$$
$$17$$$$1 + 6 T + 26 T^{2} + 102 T^{3} + 289 T^{4}$$
$$19$$$$1 + T + 10 T^{2} + 19 T^{3} + 361 T^{4}$$
$$23$$$$1 - 5 T + 17 T^{2} - 115 T^{3} + 529 T^{4}$$
$$29$$$$1 + 6 T + 10 T^{2} + 174 T^{3} + 841 T^{4}$$
$\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.