# Properties

 Label 324.a Sato-Tate group $E_3$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 324.a

Label Equation
324.a.648.1 $$y^2 + (x^3 + x + 1)y = x^5 + 2x^4 + 2x^3 + x^2$$

## L-function data

Analytic rank:$$0$$

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

Good L-factors:
Prime L-Factor
$$5$$$$1 - 5 T^{2} + 25 T^{4}$$
$$7$$$$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$
$$11$$$$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$$17$$$$( 1 + 3 T + 17 T^{2} )^{2}$$
$$19$$$$( 1 + T + 19 T^{2} )^{2}$$
$$23$$$$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$
$$29$$$$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $E_3$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$

## Endomorphisms of the Jacobian

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

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial $$x^{3} - 3 x - 1$$

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.