# Properties

 Label 576.a Sato-Tate group $E_2$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 576.a

Label Equation
576.a.576.1 $$y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x$$

## L-function data

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

Prime L-Factor
$$2$$$$1 + 2 T + 2 T^{2}$$
$$3$$$$1 + T^{2}$$

Good L-factors:
Prime L-Factor
$$5$$$$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$7$$$$( 1 + 2 T + 7 T^{2} )^{2}$$
$$11$$$$( 1 - 11 T^{2} )^{2}$$
$$13$$$$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$17$$$$( 1 + 2 T + 17 T^{2} )^{2}$$
$$19$$$$1 - 22 T^{2} + 361 T^{4}$$
$$23$$$$( 1 - 4 T + 23 T^{2} )^{2}$$
$$29$$$$1 - 22 T^{2} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $E_2$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

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

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

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

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