# Properties

 Label 450.a Conductor $450$ 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 450.a

Label Equation
450.a.2700.1 $$y^2 + (x^3 + 1)y = x^5 + 3x^4 + 3x^3 + 3x^2 + x$$
450.a.36450.1 $$y^2 + (x^3 + 1)y = x^5 - 4x^4 - 9x^3 + 28x^2 - 6x - 16$$

## L-function data

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

Prime L-Factor
$$2$$$$( 1 + T )( 1 + T + 2 T^{2} )$$
$$3$$$$( 1 - T )( 1 + T )$$
$$5$$$$( 1 - T )( 1 + T )$$

Good L-factors:
Prime L-Factor
$$7$$$$( 1 + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$( 1 + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$$13$$$$( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$( 1 - 6 T + 17 T^{2} )( 1 - 2 T + 17 T^{2} )$$
$$19$$$$( 1 - 4 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )$$
$$23$$$$( 1 + 23 T^{2} )^{2}$$
$$29$$$$( 1 + 2 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$$
$\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 15.a
Elliptic curve isogeny class 30.a

## 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.