Properties

 Label 100352.d Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

Related objects

Genus 2 curves in isogeny class 100352.d

Label Equation
100352.d.100352.1 $$y^2 = x^5 - x^4 - 4x^3 + 3x^2 + 4x - 2$$

L-function data

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

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

Good L-factors:
Prime L-Factor
$$3$$$$1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$$5$$$$( 1 - T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$
$$11$$$$( 1 - 4 T + 11 T^{2} )( 1 + 3 T + 11 T^{2} )$$
$$13$$$$1 - 2 T^{2} + 169 T^{4}$$
$$17$$$$( 1 - 2 T + 17 T^{2} )( 1 + 7 T + 17 T^{2} )$$
$$19$$$$1 - 5 T + 34 T^{2} - 95 T^{3} + 361 T^{4}$$
$$23$$$$1 + T + 6 T^{2} + 23 T^{3} + 529 T^{4}$$
$$29$$$$1 - 4 T + 6 T^{2} - 116 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

Sato-Tate group

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$

Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

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

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.