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

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

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x )^{2}( 1 - 4 x + 9 x^{2} - 16 x^{3} + 16 x^{4} )$
	$1 - 8 x + 29 x^{2} - 68 x^{3} + 116 x^{4} - 128 x^{5} + 64 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.117169895439$, $\pm0.478661301576$
Angle rank:	$2$ (numerical)

This isogeny class is not simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$2484$	$193158$	$12916800$	$1005365526$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$11$	$45$	$191$	$957$	$4163$	$16461$	$65087$	$261549$	$1049891$

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 $\F_{2^{2}}$.

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

The isogeny class factors as 1.4.ae $\times$ 2.4.ae_j 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 $\infty$.
• 2.4.ae_j : 4.0.4752.1.

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.a_ad_ae	$2$	3.16.ag_ap_hc
3.4.a_ad_e	$2$	3.16.ag_ap_hc
3.4.i_bd_cq	$2$	3.16.ag_ap_hc
3.4.ac_f_ao	$3$	(not in LMFDB)