Abelian variety isogeny class 2.2.a_b over $\F_{2}$

• Label: 2.2.a_b
• Base field: $\F_{2}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: Yes
• Supersingular: No
• Simple: Yes
• Geometrically simple: No
• Primitive: Yes
• Principally polarizable: Yes
• Contains a Jacobian: Yes

Invariants

Base field:	$\F_{2}$
Dimension:	$2$
L-polynomial:	$1 + x^{2} + 4 x^{4}$
Frobenius angles:	$\pm0.290215311628$, $\pm0.709784688372$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{3}, \sqrt{-5})\)
Galois group:	$C_2^2$
Jacobians:	1

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2+(x^2+x+1)y=x^5+x^3+x^2+1$

Point counts of the abelian variety

$r$	1	2	3	4	5	6	7	8	9	10
$A(\F_{q^r})$	6	36	54	576	1086	2916	16134	57600	262926	1179396

Point counts of the curve

$r$	1	2	3	4	5	6	7	8	9	10
$C(\F_{q^r})$	3	7	9	31	33	43	129	223	513	1147

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{3}, \sqrt{-5})\).

Endomorphism algebra over $\overline{\F}_{2}$

The base change of $A$ to $\F_{2^{2}}$ is 1.4.b^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

All geometric endomorphisms are defined over $\F_{2^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension Degree	Common base change
2.2.a_ab	$4$	2.16.o_dd
2.2.ad_f	$12$	(not in LMFDB)
2.2.d_f	$12$	(not in LMFDB)