# Properties

 Label 762.a Conductor $762$ 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 762.a

Label Equation
762.a.3048.1 $$y^2 + (x^3 + x^2 + x)y = x^2 + x + 1$$
762.a.82296.1 $$y^2 + (x^2 + x)y = x^5 - 8x^4 + 14x^3 + 2x^2 - x$$

## L-function data

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

Bad L-factors:
Prime L-Factor
$$2$$$$( 1 - T )( 1 + T + 2 T^{2} )$$
$$3$$$$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$127$$$$( 1 + T )( 1 + 16 T + 127 T^{2} )$$

Good L-factors:
Prime L-Factor
$$5$$$$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$
$$7$$$$1 - T + 6 T^{2} - 7 T^{3} + 49 T^{4}$$
$$11$$$$1 + 3 T + 10 T^{2} + 33 T^{3} + 121 T^{4}$$
$$13$$$$1 + 2 T - 6 T^{2} + 26 T^{3} + 169 T^{4}$$
$$17$$$$1 - T + 16 T^{2} - 17 T^{3} + 289 T^{4}$$
$$19$$$$1 - 5 T + 26 T^{2} - 95 T^{3} + 361 T^{4}$$
$$23$$$$1 - T - 2 T^{2} - 23 T^{3} + 529 T^{4}$$
$$29$$$$( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$$
$\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.