Abelian variety isogeny class 3.8.ak_by_agj over $\F_{2^{3}}$

• Label: 3.8.ak_by_agj
• Base field: $\F_{2^{3}}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{3}}$
Dimension:	$3$
L-polynomial:	$( 1 - 5 x + 8 x^{2} )( 1 - 5 x + 17 x^{2} - 40 x^{3} + 64 x^{4} )$
	$1 - 10 x + 50 x^{2} - 165 x^{3} + 400 x^{4} - 640 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.154919815756$, $\pm0.178413517577$, $\pm0.488253149089$
Angle rank:	$1$ (numerical)
Isomorphism classes:	4

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$148$	$263144$	$136308592$	$68719003024$	$35779916211908$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$65$	$518$	$4097$	$33319$	$265142$	$2102743$	$16777217$	$134210174$	$1073741825$

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 $\F_{2^{18}}$.

Endomorphism algebra over $\F_{2^{3}}$

The isogeny class factors as 1.8.af $\times$ 2.8.af_r and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.8.af : \(\Q(\sqrt{-7}) \).
• 2.8.af_r : \(\Q(\sqrt{-3}, \sqrt{-7})\).

Endomorphism algebra over $\overline{\F}_{2^{3}}$

The base change of $A$ to $\F_{2^{18}}$ is 1.262144.bml^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj $\times$ 2.64.j_r. The endomorphism algebra for each factor is:

  • 1.64.aj : \(\Q(\sqrt{-7}) \).
  • 2.64.j_r : \(\Q(\sqrt{-3}, \sqrt{-7})\).
• Endomorphism algebra over $\F_{2^{9}}$
  The base change of $A$ to $\F_{2^{9}}$ is 1.512.af $\times$ 1.512.f^2. The endomorphism algebra for each factor is:

  • 1.512.af : \(\Q(\sqrt{-7}) \).
  • 1.512.f^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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
3.8.a_a_af	$2$	(not in LMFDB)
3.8.a_a_f	$2$	(not in LMFDB)
3.8.k_by_gj	$2$	(not in LMFDB)
3.8.f_ab_abt	$3$	(not in LMFDB)