Genus 2 isogeny class 114075.b

• Label: 114075.b
• Conductor: $114075$
• 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 114075.b

Label	Equation
114075.b.570375.1	$y^2 + (x^3 + x^2 + 1)y = -2x^4 - 5x^3 + 5x + 1$

L-function data

Analytic rank:

$1$

Mordell-Weil rank:

$1$

Bad L-factors:

Prime	L-Factor
$3$	$1 - T + 3 T^{2}$
$5$	$( 1 - T )( 1 + T )$
$13$	$1 - 4 T + 13 T^{2}$

Good L-factors:

Prime	L-Factor	Isogeny Class over $\mathbb{F}_p$
$2$	$1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}$	2.2.ab_d
$7$	$1 - T + 10 T^{2} - 7 T^{3} + 49 T^{4}$	2.7.ab_k
$11$	$( 1 - T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$	2.11.f_q
$17$	$( 1 - 5 T + 17 T^{2} )^{2}$	2.17.ak_ch
$19$	$( 1 + 3 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$	2.19.k_ch
$23$	$1 + 2 T - 5 T^{2} + 46 T^{3} + 529 T^{4}$	2.23.c_af
$29$	$( 1 + 29 T^{2} )( 1 + 9 T + 29 T^{2} )$	2.29.j_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.