Genus 2 isogeny class 196.a

• Label: 196.a
• Conductor: $196$
• Sato-Tate group: $E_1$
• End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\mathrm{M}_2(\R)$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\mathrm{M}_2(\Q)$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\mathrm{M}_2(\Q)$
• Q‾\overline{\Q}-simple: no
• GL2\mathrm{GL}_2-type: no

Genus 2 curves in isogeny class 196.a

Label	Equation
196.a.21952.1	$y^2 + (x^2 + x)y = x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1$

L-function data

Analytic rank:

$0$

Mordell-Weil rank:

$0$

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over $\mathbb{F}_p$
$3$	$( 1 + 2 T + 3 T^{2} )^{2}$	2.3.e_k
$5$	$( 1 + 5 T^{2} )^{2}$	2.5.a_k
$11$	$( 1 + 11 T^{2} )^{2}$	2.11.a_w
$13$	$( 1 + 4 T + 13 T^{2} )^{2}$	2.13.i_bq
$17$	$( 1 - 6 T + 17 T^{2} )^{2}$	2.17.am_cs
$19$	$( 1 - 2 T + 19 T^{2} )^{2}$	2.19.ae_bq
$23$	$( 1 + 23 T^{2} )^{2}$	2.23.a_bu
$29$	$( 1 + 6 T + 29 T^{2} )^{2}$	2.29.m_dq
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

$\mathrm{ST} =$ $E_1$, $\quad \mathrm{ST}^0 = \mathrm{SU}(2)$

Decomposition of the Jacobian

Splits over $\Q$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 14.a

Endomorphisms of the Jacobian

Not of $\GL_2$-type over $\Q$

Endomorphism algebra over $\Q$:

$\End (J_{}) \otimes \Q$	$\simeq$	$\mathrm{M}_2($$\Q$$)$
$\End (J_{}) \otimes \R$	$\simeq$	$\mathrm{M}_2 (\R)$

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.