# Properties

 Label 11664.a Sato-Tate group $D_{6,2}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\C)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\mathrm{CM})$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 11664.a

Label Equation
11664.a.11664.1 $$y^2 + y = -x^6$$

## L-function data

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

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

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

## Sato-Tate group

$$\mathrm{ST} =$$ $D_{6,2}$, $$\quad \mathrm{ST}^0 = \mathrm{U}(1)$$

## 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 $$K = \Q (a)$$ with defining polynomial $$x^{12} - 3 x^{10} - 8 x^{9} - 6 x^{8} + 12 x^{7} + 47 x^{6} + 78 x^{5} + 78 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 1$$

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.