# Properties

 Label 3264.a Sato-Tate group $N(G_{3,3})$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

## Genus 2 curves in isogeny class 3264.a

Label Equation
3264.a.3264.1 $$y^2 + x^3y = x^5 - 3x^3 + 4x - 2$$

## L-function data

Analytic rank:$$1$$

Prime L-Factor
$$2$$$$1 + 2 T + 2 T^{2}$$
$$3$$$$( 1 + T )( 1 + 2 T + 3 T^{2} )$$
$$17$$$$( 1 + T )( 1 + 6 T + 17 T^{2} )$$

Good L-factors:
Prime L-Factor
$$5$$$$1 + 2 T^{2} + 25 T^{4}$$
$$7$$$$1 + 2 T^{2} + 49 T^{4}$$
$$11$$$$( 1 + 2 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$$13$$$$1 - 6 T^{2} + 169 T^{4}$$
$$19$$$$( 1 + 19 T^{2} )( 1 + 4 T + 19 T^{2} )$$
$$23$$$$1 + 6 T^{2} + 529 T^{4}$$
$$29$$$$1 - 46 T^{2} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

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

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