# Properties

 Label 862.a Conductor $862$ 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 862.a

Label Equation
862.a.862.1 $$y^2 + (x^3 + 1)y = x^5 - 2x^4 - 7x^3 + 7x^2 + 2x + 5$$
862.a.6896.1 $$y^2 + (x^2 + x)y = 4x^5 + 6x^4 - 3x^2 - x$$

## L-function data

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

Bad L-factors:
Prime L-Factor
$$2$$$$( 1 - T )( 1 + T + 2 T^{2} )$$
$$431$$$$( 1 - T )( 1 - 16 T + 431 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$$5$$$$1 + T + 5 T^{3} + 25 T^{4}$$
$$7$$$$1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4}$$
$$11$$$$( 1 - T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$$13$$$$( 1 - 6 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$( 1 - 2 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )$$
$$19$$$$( 1 - 4 T + 19 T^{2} )( 1 + T + 19 T^{2} )$$
$$23$$$$1 - 5 T + 38 T^{2} - 115 T^{3} + 529 T^{4}$$
$$29$$$$1 - 3 T - 16 T^{2} - 87 T^{3} + 841 T^{4}$$
$\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.