Abelian variety isogeny class 2.64.az_km over $\F_{2^{6}}$

• Label: 2.64.az_km
• Base field: $\F_{2^{6}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{6}}$
Dimension:	$2$
L-polynomial:	$( 1 - 8 x )^{2}( 1 - 9 x + 64 x^{2} )$
	$1 - 25 x + 272 x^{2} - 1600 x^{3} + 4096 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.309839631512$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2744$	$16447536$	$68712424424$	$281437900380000$	$1152840305690007704$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$4016$	$262120$	$16775008$	$1073666200$	$68718478736$	$4398038699080$	$281474940914368$	$18014398452403960$	$1152921504505059056$

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 $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2^{6}}$

The isogeny class factors as 1.64.aq $\times$ 1.64.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.64.aq : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.64.aj : \(\Q(\sqrt{-7}) \).

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
2.64.ah_aq	$2$	(not in LMFDB)
2.64.h_aq	$2$	(not in LMFDB)
2.64.z_km	$2$	(not in LMFDB)
2.64.ab_ce	$3$	(not in LMFDB)