Abelian variety isogeny class 2.16.aj_ca over $\F_{2^{4}}$

• Label: 2.16.aj_ca
• Base field: $\F_{2^{4}}$
• 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:	$\F_{2^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 16 x^{2} )( 1 - 4 x + 16 x^{2} )$
	$1 - 9 x + 52 x^{2} - 144 x^{3} + 256 x^{4}$
Frobenius angles:	$\pm0.285098958592$, $\pm0.333333333333$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$156$	$72072$	$17795700$	$4342338000$	$1098937569756$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$280$	$4340$	$66256$	$1048028$	$16763992$	$268386308$	$4294949536$	$68720108780$	$1099514547880$

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 $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2^{4}}$

The isogeny class factors as 1.16.af $\times$ 1.16.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.16.af : \(\Q(\sqrt{-39}) \).
• 1.16.ae : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{2^{4}}$

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.el $\times$ 1.4096.ey. The endomorphism algebra for each factor is:

• 1.4096.el : \(\Q(\sqrt{-39}) \).
• 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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
2.16.ab_m	$2$	2.256.x_ya
2.16.b_m	$2$	2.256.x_ya
2.16.j_ca	$2$	2.256.x_ya
2.16.d_ai	$3$	(not in LMFDB)