Genus 2 isogeny class 644.a

• Label: 644.a
• Conductor: $644$
• 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 644.a

Label	Equation
644.a.2576.1	$y^2 + (x^2 + x)y = -5x^6 + 11x^5 - 20x^4 + 20x^3 - 20x^2 + 11x - 5$
644.a.659456.1	$y^2 + (x^2 + x)y = -3x^6 - 13x^5 + 4x^4 + 51x^3 + 4x^2 - 13x - 3$

L-function data

Analytic rank:

$0$

Mordell-Weil rank:

$0$

Bad L-factors:

Prime	L-Factor
$2$	$( 1 + T )^{2}$
$7$	$( 1 - T )( 1 + 4 T + 7 T^{2} )$
$23$	$( 1 - T )( 1 + 23 T^{2} )$

Good L-factors:

Prime	L-Factor
$3$	$( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$
$5$	$( 1 - 4 T + 5 T^{2} )( 1 + 5 T^{2} )$
$11$	$( 1 - 2 T + 11 T^{2} )( 1 + 11 T^{2} )$
$13$	$( 1 + 2 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$
$17$	$( 1 - 6 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )$
$19$	$( 1 - 2 T + 19 T^{2} )( 1 + 2 T + 19 T^{2} )$
$29$	$( 1 - 2 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$
$\cdots$	$\cdots$

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 14.a
  Elliptic curve isogeny class 46.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.