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

• Label: 3.8.ao_dl_amm
• Base field: $\F_{2^{3}}$
• Dimension: $3$
• $p$-rank: $2$
• 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:	$( 1 - 4 x + 8 x^{2} )( 1 - 5 x + 8 x^{2} )^{2}$
	$1 - 14 x + 89 x^{2} - 324 x^{3} + 712 x^{4} - 896 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.154919815756$, $\pm0.154919815756$, $\pm0.250000000000$
Angle rank:	$1$ (numerical)

This isogeny class is not simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$80$	$203840$	$140644880$	$72554809600$	$36060193536400$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$47$	$535$	$4319$	$33575$	$264143$	$2100695$	$16780991$	$134216455$	$1073721647$

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.af^2 $\times$ 1.8.ae 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^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.8.ae : \(\Q(\sqrt{-1}) \).

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

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

• 1.4096.bv^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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^2 $\times$ 1.64.a. The endomorphism algebra for each factor is:

  • 1.64.aj^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
  • 1.64.a : \(\Q(\sqrt{-1}) \).

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_ab_ag
$\F_{2}$	3.2.e_l_s

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.8.ag_j_e	$2$	(not in LMFDB)
3.8.ae_ab_bk	$2$	(not in LMFDB)
3.8.e_ab_abk	$2$	(not in LMFDB)
3.8.g_j_ae	$2$	(not in LMFDB)
3.8.o_dl_mm	$2$	(not in LMFDB)
3.8.b_f_m	$3$	(not in LMFDB)