Genus 2 isogeny class 10137.a

• Label: 10137.a
• Conductor: $10137$
• 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 10137.a

Label	Equation
10137.a.10137.1	$y^2 + (x^3 + x + 1)y = -2x^4 + 4x^2 - x - 4$

L-function data

Analytic rank:

$1$

Mordell-Weil rank:

$1$

Bad L-factors:

Prime	L-Factor
$3$	$( 1 + T )( 1 + 3 T^{2} )$
$31$	$( 1 - T )( 1 + 3 T + 31 T^{2} )$
$109$	$( 1 + T )( 1 + 9 T + 109 T^{2} )$

Good L-factors:

Prime	L-Factor	Isogeny Class over $\mathbb{F}_p$
$2$	$1 + T + T^{2} + 2 T^{3} + 4 T^{4}$	2.2.b_b
$5$	$1 - T + 2 T^{2} - 5 T^{3} + 25 T^{4}$	2.5.ab_c
$7$	$( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$	2.7.e_j
$11$	$( 1 - 4 T + 11 T^{2} )( 1 + 2 T + 11 T^{2} )$	2.11.ac_o
$13$	$1 - 2 T + 10 T^{2} - 26 T^{3} + 169 T^{4}$	2.13.ac_k
$17$	$1 + 5 T + 22 T^{2} + 85 T^{3} + 289 T^{4}$	2.17.f_w
$19$	$1 + 3 T - 6 T^{2} + 57 T^{3} + 361 T^{4}$	2.19.d_ag
$23$	$1 + 3 T - 2 T^{2} + 69 T^{3} + 529 T^{4}$	2.23.d_ac
$29$	$( 1 - 8 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$	2.29.c_aw
$\cdots$	$\cdots$	$\cdots$

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.