Genus 2 isogeny class 75625.b

• Label: 75625.b
• Conductor: $75625$
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\R \times \R$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\mathsf{RM}$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\mathsf{RM}$
• Q‾\overline{\Q}-simple: yes
• GL2\mathrm{GL}_2-type: yes

Genus 2 curves in isogeny class 75625.b

Label	Equation
75625.b.75625.1	$y^2 + (x^3 + x + 1)y = x^5 - x^4 - 6x^3 + 11x - 7$

L-function data

Analytic rank:

$2$  (upper bound)

Mordell-Weil rank:

$2$

Bad L-factors:

Prime	L-Factor
$5$	$1$
$11$	$( 1 + T )^{2}$

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
$3$	$1 + T + 3 T^{2} + 3 T^{3} + 9 T^{4}$	2.3.b_d
$7$	$1 + 5 T + 17 T^{2} + 35 T^{3} + 49 T^{4}$	2.7.f_r
$13$	$( 1 + 5 T + 13 T^{2} )^{2}$	2.13.k_bz
$17$	$1 + 3 T + 7 T^{2} + 51 T^{3} + 289 T^{4}$	2.17.d_h
$19$	$( 1 + T + 19 T^{2} )^{2}$	2.19.c_bn
$23$	$1 - 11 T + 73 T^{2} - 253 T^{3} + 529 T^{4}$	2.23.al_cv
$29$	$1 + 9 T + 49 T^{2} + 261 T^{3} + 841 T^{4}$	2.29.j_bx
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

$\mathrm{ST} =$ $\mathrm{SU}(2)\times\mathrm{SU}(2)$, $\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)$

Decomposition of the Jacobian

Simple over $\overline{\Q}$

Endomorphisms of the Jacobian

Of $\GL_2$-type over $\Q$

Endomorphism algebra over $\Q$:

$\End (J_{}) \otimes \Q$	$\simeq$	$\Q(\sqrt{13})$
$\End (J_{}) \otimes \R$	$\simeq$	$\R \times \R$

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.