Genus 2 isogeny class 784.a

• Label: 784.a
• Conductor: $784$
• 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: $\Q \times \Q$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\Q \times \Q$
• Q‾\overline{\Q}-simple: no
• GL2\mathrm{GL}_2-type: yes

Genus 2 curves in isogeny class 784.a

Label	Equation
784.a.1568.1	$y^2 + (x^3 + x)y = -2x^4 + 3x^2 - 2$
784.a.43904.1	$y^2 + (x^3 + x)y = 4x^4 + 27x^2 + 56$

L-function data

Analytic rank:

$0$

Mordell-Weil rank:

$0$

Bad L-factors:

Prime	L-Factor
$2$	$1 + T$
$7$	$( 1 - T )( 1 + T )$

Good L-factors:

Prime	L-Factor	Isogeny Class over $\mathbb{F}_p$
$3$	$( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$	2.3.c_g
$5$	$( 1 - 2 T + 5 T^{2} )( 1 + 5 T^{2} )$	2.5.ac_k
$11$	$( 1 + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$	2.11.e_w
$13$	$( 1 - 2 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$	2.13.c_s
$17$	$( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )$	2.17.a_ac
$19$	$( 1 - 8 T + 19 T^{2} )( 1 - 2 T + 19 T^{2} )$	2.19.ak_cc
$23$	$( 1 + 23 T^{2} )^{2}$	2.23.a_bu
$29$	$( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$	2.29.a_w
$\cdots$	$\cdots$	$\cdots$

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

Splits over $\Q$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 56.a
  Elliptic curve isogeny class 14.a

Endomorphisms of the Jacobian

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

Endomorphism algebra over $\Q$:

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

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.