# Properties

 Label 6437.a Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Learn more about

## Genus 2 curves in isogeny class 6437.a

Label Equation
6437.a.6437.1 $$y^2 + (x^3 + x^2 + x + 1)y = x^3 - 2x$$

## L-function data

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

Bad L-factors:
Prime L-Factor
$$41$$$$( 1 + T )( 1 - T + 41 T^{2} )$$
$$157$$$$( 1 + T )( 1 - 18 T + 157 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$( 1 + T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )$$
$$7$$$$1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4}$$
$$11$$$$1 + 4 T + 15 T^{2} + 44 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 4 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$$17$$$$1 - 2 T + 15 T^{2} - 34 T^{3} + 289 T^{4}$$
$$19$$$$1 + 4 T + 29 T^{2} + 76 T^{3} + 361 T^{4}$$
$$23$$$$1 + 23 T^{2} + 529 T^{4}$$
$$29$$$$1 - T - 5 T^{2} - 29 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

See L-function page for more information

## Sato-Tate group

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$

## 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.