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

• Label: 3.16.au_gv_abiu
• Base field: $\F_{2^{4}}$
• Dimension: $3$
• $p$-rank: $2$
• 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:	$3$
L-polynomial:	$( 1 - 4 x )^{2}( 1 - 12 x + 65 x^{2} - 192 x^{3} + 256 x^{4} )$
	$1 - 20 x + 177 x^{2} - 904 x^{3} + 2832 x^{4} - 5120 x^{5} + 4096 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.0826163580681$, $\pm0.320878822416$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1062$	$13965300$	$67160711142$	$279547543828800$	$1149343372460251782$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$211$	$4005$	$65087$	$1045317$	$16760947$	$268387221$	$4294927103$	$68719807701$	$1099512984691$

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 $\F_{2^{4}}$.

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

The isogeny class factors as 1.16.ai $\times$ 2.16.am_cn 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.am_cn : 4.0.27792.2.

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.ae_ap_fg	$2$	(not in LMFDB)
3.16.e_ap_afg	$2$	(not in LMFDB)
3.16.u_gv_biu	$2$	(not in LMFDB)
3.16.ai_bh_aeu	$3$	(not in LMFDB)