Abelian variety isogeny class 4.4.ak_bt_aex_kq over $\F_{2^{2}}$

• Label: 4.4.ak_bt_aex_kq
• Base field: $\F_{2^{2}}$
• Dimension: $4$
• $p$-rank: $3$
• 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:	$4$
L-polynomial:	$( 1 - 2 x )^{2}( 1 - 6 x + 17 x^{2} - 35 x^{3} + 68 x^{4} - 96 x^{5} + 64 x^{6} )$
	$1 - 10 x + 45 x^{2} - 127 x^{3} + 276 x^{4} - 508 x^{5} + 720 x^{6} - 640 x^{7} + 256 x^{8}$
Frobenius angles:	$0$, $0$, $\pm0.125725122630$, $\pm0.183009650264$, $\pm0.584457690022$
Angle rank:	$3$ (numerical)

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

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13$	$33579$	$10442341$	$3889287675$	$1192268468053$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$7$	$34$	$231$	$1110$	$4180$	$16284$	$65863$	$262303$	$1044372$

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 $\times$ 3.4.ag_r_abj 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 $\infty$.
• 3.4.ag_r_abj : 6.0.400967.1.

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
4.4.ac_ad_j_ae	$2$	(not in LMFDB)
4.4.c_ad_aj_ae	$2$	(not in LMFDB)
4.4.k_bt_ex_kq	$2$	(not in LMFDB)
4.4.ae_j_az_co	$3$	(not in LMFDB)