Abelian variety isogeny class 3.4.ag_v_abw over $\F_{2^{2}}$

• Label: 3.4.ag_v_abw
• Base field: $\F_{2^{2}}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 + 4 x^{2} )( 1 - 3 x + 4 x^{2} )^{2}$
	$1 - 6 x + 21 x^{2} - 48 x^{3} + 84 x^{4} - 96 x^{5} + 64 x^{6}$
Frobenius angles:	$\pm0.230053456163$, $\pm0.230053456163$, $\pm0.5$
Angle rank:	$1$ (numerical)

This isogeny class is not simple, not primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20$	$6400$	$355940$	$18662400$	$1199992100$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$23$	$83$	$287$	$1139$	$4319$	$16211$	$64127$	$260147$	$1048223$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2^{2}}$

The isogeny class factors as 1.4.ad^2 $\times$ 1.4.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.4.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.4.a : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab^2 $\times$ 1.16.i. The endomorphism algebra for each factor is:

• 1.16.ab^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

Subfield	Primitive Model
$\F_{2}$	3.2.ac_ab_g
$\F_{2}$	3.2.c_ab_ag

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.4.a_d_a	$2$	3.16.g_bh_hs
3.4.g_v_bw	$2$	3.16.g_bh_hs
3.4.d_j_y	$3$	(not in LMFDB)