# Properties

 Label 6845.a Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Learn more about

## Genus 2 curves in isogeny class 6845.a

Label Equation
6845.a.6845.1 $$y^2 + x^3y = x^5 - 7x^3 - 16x^2 - 15x - 5$$

## L-function data

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

Bad L-factors:
Prime L-Factor
$$5$$$$( 1 - T )( 1 + 2 T + 5 T^{2} )$$
$$37$$$$( 1 - T )( 1 + T )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$7$$$$( 1 + T + 7 T^{2} )( 1 + 3 T + 7 T^{2} )$$
$$11$$$$( 1 + 5 T + 11 T^{2} )^{2}$$
$$13$$$$( 1 - 4 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$( 1 + 17 T^{2} )( 1 + 4 T + 17 T^{2} )$$
$$19$$$$( 1 + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$( 1 - 4 T + 23 T^{2} )( 1 - 2 T + 23 T^{2} )$$
$$29$$$$( 1 - 6 T + 29 T^{2} )( 1 - 4 T + 29 T^{2} )$$
$\cdots$$\cdots$

See L-function page for more information

## Sato-Tate group

$$\mathrm{ST} =$$ $G_{3,3}$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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.