# Properties

 Label 640.a Sato-Tate group $N(G_{1,3})$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\C \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{CM} \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 640.a

Label Equation
640.a.81920.1 $$y^2 + x^3y = 3x^4 + 13x^2 + 20$$
640.a.81920.2 $$y^2 + x^3y = -3x^4 + 13x^2 - 20$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$1$$
$$5$$$$( 1 + T )( 1 + 2 T + 5 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 7 T^{2} )$$
$$11$$$$( 1 + 11 T^{2} )^{2}$$
$$13$$$$( 1 - 6 T + 13 T^{2} )( 1 - 2 T + 13 T^{2} )$$
$$17$$$$( 1 - 2 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )$$
$$19$$$$( 1 + 19 T^{2} )( 1 + 4 T + 19 T^{2} )$$
$$23$$$$( 1 - 6 T + 23 T^{2} )( 1 + 23 T^{2} )$$
$$29$$$$( 1 - 6 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $N(G_{1,3})$, $$\quad \mathrm{ST}^0 = \mathrm{U}(1)\times\mathrm{SU}(2)$$

## Endomorphisms of the Jacobian

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

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{-1})$$ with defining polynomial $$x^{2} + 1$$

Endomorphism algebra over $$\overline{\Q}$$:
 $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q(\sqrt{-1})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \C$$

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.