# Properties

 Label 15552.c Conductor $15552$ Sato-Tate group $J(E_6)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

## Genus 2 curves in isogeny class 15552.c

Label Equation
15552.c.746496.1 $$y^2 + x^3y = 3x^3 + 8$$

## L-function data

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

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

Good L-factors:
Prime L-Factor
$$5$$$$1 - 2 T^{2} + 25 T^{4}$$
$$7$$$$( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$1 + 10 T^{2} + 121 T^{4}$$
$$13$$$$( 1 - 5 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )$$
$$19$$$$1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 6 T + 23 T^{2} )( 1 + 6 T + 23 T^{2} )$$
$$29$$$$( 1 + 29 T^{2} )^{2}$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

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

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a)$$ with defining polynomial $$x^{12} - 6 x^{10} + 44 x^{6} + 60 x^{4} + 24 x^{2} + 4$$

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.