Genus 2 isogeny class 644.b

• Label: 644.b
• Conductor: $644$
• Sato-Tate group: $\mathrm{USp}(4)$
• End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\R$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\Q$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\Q$
• Q‾\overline{\Q}-simple: yes
• GL2\mathrm{GL}_2-type: no

Genus 2 curves in isogeny class 644.b

Label	Equation
644.b.14812.1	$y^2 + (x^3 + 1)y = x^5 - x^4 - 4x^3 + 5x^2 - x - 1$

L-function data

Analytic rank:

$0$

Mordell-Weil rank:

$0$

Bad L-factors:

Prime	L-Factor
$2$	$1 + T^{2}$
$7$	$( 1 + T )( 1 + 2 T + 7 T^{2} )$
$23$	$( 1 + T )( 1 - 4 T + 23 T^{2} )$

Good L-factors:

Prime	L-Factor	Isogeny Class over $\mathbb{F}_p$
$3$	$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$	2.3.c_c
$5$	$1 + 4 T^{2} + 25 T^{4}$	2.5.a_e
$11$	$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$	2.11.ac_c
$13$	$( 1 - 4 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$	2.13.ac_s
$17$	$1 + 2 T - 6 T^{2} + 34 T^{3} + 289 T^{4}$	2.17.c_ag
$19$	$1 - 12 T^{2} + 361 T^{4}$	2.19.a_am
$29$	$( 1 - 8 T + 29 T^{2} )( 1 + 29 T^{2} )$	2.29.ai_cg
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

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

Decomposition of the Jacobian

Simple over $\overline{\Q}$

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.