Abelian variety isogeny class 3.4.ag_n_au over $\F_{2^{2}}$

• Label: 3.4.ag_n_au
• Base field: $\F_{2^{2}}$
• 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:	$\F_{2^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x )^{2}( 1 - 2 x + x^{2} - 8 x^{3} + 16 x^{4} )$
	$1 - 6 x + 13 x^{2} - 20 x^{3} + 52 x^{4} - 96 x^{5} + 64 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.0935673124239$, $\pm0.651114279890$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$2016$	$132104$	$14716800$	$1046805768$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$7$	$23$	$223$	$999$	$3871$	$16295$	$65791$	$261095$	$1048607$

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 $\times$ 2.4.ac_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 $\infty$.
• 2.4.ac_b : 4.0.1088.2.

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
3.4.ac_ad_m	$2$	3.16.ak_bh_acu
3.4.c_ad_am	$2$	3.16.ak_bh_acu
3.4.g_n_u	$2$	3.16.ak_bh_acu
3.4.a_b_ao	$3$	(not in LMFDB)