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

• Label: 3.8.al_ci_aia
• Base field: $\F_{2^{3}}$
• Dimension: $3$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{3}}$
Dimension:	$3$
L-polynomial:	$( 1 - 4 x + 8 x^{2} )( 1 - 7 x + 24 x^{2} - 56 x^{3} + 64 x^{4} )$
	$1 - 11 x + 60 x^{2} - 208 x^{3} + 480 x^{4} - 704 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.0585111942353$, $\pm0.250000000000$, $\pm0.418160225599$
Angle rank:	$2$ (numerical)

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

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$130$	$256880$	$140807290$	$68157970400$	$34851402248650$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$64$	$538$	$4064$	$32458$	$261712$	$2097114$	$16772032$	$134186218$	$1073703984$

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 $\F_{2^{3}}$

The isogeny class factors as 1.8.ae $\times$ 2.8.ah_y 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.ae : \(\Q(\sqrt{-1}) \).
• 2.8.ah_y : 4.0.2312.1.

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey $\times$ 2.4096.agf_vki. The endomorphism algebra for each factor is:

• 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.4096.agf_vki : 4.0.2312.1.

Remainder of endomorphism lattice by field

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

  • 1.64.a : \(\Q(\sqrt{-1}) \).
  • 2.64.ab_adc : 4.0.2312.1.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.

Subfield	Primitive Model
$\F_{2}$	3.2.b_a_ae

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.8.ad_e_aq	$2$	(not in LMFDB)
3.8.d_e_q	$2$	(not in LMFDB)
3.8.l_ci_ia	$2$	(not in LMFDB)