Abelian variety isogeny class 2.4.ad_e over $\F_{2^{2}}$

• Label: 2.4.ad_e
• Base field: $\F_{2^{2}}$
• 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:	$\F_{2^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 2 x )^{2}( 1 + x + 4 x^{2} )$
	$1 - 3 x + 4 x^{2} - 12 x^{3} + 16 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.580430623255$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$216$	$2646$	$54000$	$1043646$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$16$	$38$	$208$	$1022$	$3976$	$15878$	$65248$	$261902$	$1044856$

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 $\times$ 1.4.b 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.b : \(\Q(\sqrt{-15}) \).

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
2.4.af_m	$2$	2.16.ab_ay
2.4.d_e	$2$	2.16.ab_ay
2.4.f_m	$2$	2.16.ab_ay
2.4.d_k	$3$	2.64.abb_ls