Abelian variety isogeny class 3.16.ar_fh_abae over $\F_{2^{4}}$

• Label: 3.16.ar_fh_abae
• Base field: $\F_{2^{4}}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{2^{4}}$
Dimension:	$3$
L-polynomial:	$( 1 - 4 x )^{2}( 1 - 9 x + 49 x^{2} - 144 x^{3} + 256 x^{4} )$
	$1 - 17 x + 137 x^{2} - 680 x^{3} + 2192 x^{4} - 4352 x^{5} + 4096 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.211195784157$, $\pm0.390536017683$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1377$	$15801075$	$69278307588$	$280790174895075$	$1150658848649701017$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$242$	$4131$	$65378$	$1046520$	$16770623$	$268426872$	$4294883906$	$68718429027$	$1099505556722$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{4}}$.

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

The isogeny class factors as 1.16.ai $\times$ 2.16.aj_bx 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.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.16.aj_bx : 4.0.2873.1.

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
3.16.ab_ah_ea	$2$	(not in LMFDB)
3.16.b_ah_aea	$2$	(not in LMFDB)
3.16.r_fh_bae	$2$	(not in LMFDB)
3.16.af_bd_ado	$3$	(not in LMFDB)