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

• Label: 3.4.af_k_aq
• Base field: $\F_{2^{2}}$
• Dimension: $3$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x )^{2}( 1 - 3 x + 4 x^{2} )( 1 + 2 x + 4 x^{2} )$
	$1 - 5 x + 10 x^{2} - 16 x^{3} + 40 x^{4} - 80 x^{5} + 64 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.230053456163$, $\pm0.666666666667$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14$	$3024$	$177674$	$17690400$	$1099070714$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$12$	$42$	$272$	$1050$	$3888$	$16170$	$64832$	$259098$	$1046352$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{6}}$.

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

The isogeny class factors as 1.4.ae $\times$ 1.4.ad $\times$ 1.4.c 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.ae : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4.ad : \(\Q(\sqrt{-7}) \).
• 1.4.c : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq^2 $\times$ 1.64.j. The endomorphism algebra for each factor is:

• 1.64.aq^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.64.j : \(\Q(\sqrt{-7}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.4.aj_bm_ads	$2$	3.16.af_u_aey
3.4.ad_c_a	$2$	3.16.af_u_aey
3.4.ab_ac_q	$2$	3.16.af_u_aey
3.4.b_ac_aq	$2$	3.16.af_u_aey
3.4.d_c_a	$2$	3.16.af_u_aey
3.4.f_k_q	$2$	3.16.af_u_aey
3.4.j_bm_ds	$2$	3.16.af_u_aey
3.4.al_ca_afg	$3$	(not in LMFDB)
3.4.b_e_ae	$3$	(not in LMFDB)