Genus 2 isogeny class 971.a

• Label: 971.a
• Conductor: $971$
• 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 971.a

Label	Equation
971.a.971.1	$y^2 + y = x^5 - 2x^3 + x$

L-function data

Analytic rank:

$1$

Mordell-Weil rank:

$1$

Bad L-factors:

Prime	L-Factor
$971$	$( 1 - T )( 1 + 45 T + 971 T^{2} )$

Good L-factors:

Prime	L-Factor	Isogeny Class over $\mathbb{F}_p$
$2$	$( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$	2.2.c_e
$3$	$1 + 3 T + 5 T^{2} + 9 T^{3} + 9 T^{4}$	2.3.d_f
$5$	$1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4}$	2.5.d_l
$7$	$1 - T + T^{2} - 7 T^{3} + 49 T^{4}$	2.7.ab_b
$11$	$1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4}$	2.11.f_o
$13$	$1 - T - 8 T^{2} - 13 T^{3} + 169 T^{4}$	2.13.ab_ai
$17$	$1 + 2 T + 2 T^{2} + 34 T^{3} + 289 T^{4}$	2.17.c_c
$19$	$1 - T + 3 T^{2} - 19 T^{3} + 361 T^{4}$	2.19.ab_d
$23$	$1 - T + 35 T^{2} - 23 T^{3} + 529 T^{4}$	2.23.ab_bj
$29$	$1 + T - 18 T^{2} + 29 T^{3} + 841 T^{4}$	2.29.b_as
$\cdots$	$\cdots$	$\cdots$

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.