# Properties

 Label 12544.h Conductor $12544$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 12544.h

Label Equation
12544.h.401408.1 $$y^2 = x^5 - x^4 + 4x^3 - x^2 + x$$

## L-function data

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

Prime L-Factor
$$2$$$$1$$
$$7$$$$( 1 - T )( 1 + T )$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$11$$$$( 1 - 4 T + 11 T^{2} )( 1 + 11 T^{2} )$$
$$13$$$$( 1 - 2 T + 13 T^{2} )( 1 + 13 T^{2} )$$
$$17$$$$( 1 + 2 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )$$
$$19$$$$( 1 - 2 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$( 1 + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$
$$29$$$$( 1 - 6 T + 29 T^{2} )( 1 - 2 T + 29 T^{2} )$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 112.a
Elliptic curve isogeny class 112.b

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

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

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

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