# Properties

 Label 4489.a Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 4489.a

Label Equation
4489.a.4489.1 $$y^2 + (x^3 + x + 1)y = x^5 - x$$

## L-function data

Analytic rank:$$2$$  (upper bound)
Mordell-Weil rank:$$2$$

Prime L-Factor
$$67$$$$( 1 + T )^{2}$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$3$$$$1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4}$$
$$5$$$$( 1 + 3 T + 5 T^{2} )^{2}$$
$$7$$$$1 + T + 3 T^{2} + 7 T^{3} + 49 T^{4}$$
$$11$$$$1 + 17 T^{2} + 121 T^{4}$$
$$13$$$$1 + 7 T + 27 T^{2} + 91 T^{3} + 169 T^{4}$$
$$17$$$$1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4}$$
$$19$$$$1 - T + 27 T^{2} - 19 T^{3} + 361 T^{4}$$
$$23$$$$1 - 6 T + 35 T^{2} - 138 T^{3} + 529 T^{4}$$
$$29$$$$1 + 6 T + 47 T^{2} + 174 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{5})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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.