# Properties

 Label 20736.k Sato-Tate group $D_{2,1}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\C)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\mathrm{CM})$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 20736.k

Label Equation
20736.k.373248.1 $$y^2 + x^3y = 2$$

## L-function data

Analytic rank:$$1$$
Mordell-Weil rank:$$1$$

Prime L-Factor
$$2$$$$1$$
$$3$$$$1$$

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

## Sato-Tate group

$$\mathrm{ST} =$$ $D_{2,1}$, $$\quad \mathrm{ST}^0 = \mathrm{U}(1)$$

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

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

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ with defining polynomial $$x^{4} + 2 x^{2} + 4$$

Endomorphism algebra over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q(\sqrt{-3})$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\C)$$

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