# Properties

 Label 12544.a Conductor $12544$ Sato-Tate group $E_6$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\mathsf{CM}$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 12544.a

Label Equation
12544.a.12544.1 $$y^2 = x^5 + 2x^4 - x^3 - 3x^2 - x$$

## L-function data

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

Prime L-Factor
$$2$$$$1$$
$$7$$$$1 - 4 T + 7 T^{2}$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$$5$$$$1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4}$$
$$11$$$$1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 13 T^{2} )^{2}$$
$$17$$$$1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$1 - 15 T + 98 T^{2} - 345 T^{3} + 529 T^{4}$$
$$29$$$$( 1 + 6 T + 29 T^{2} )^{2}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $E_6$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

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

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{28})^+$$ with defining polynomial $$x^{6} - 7 x^{4} + 14 x^{2} - 7$$

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.