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

• Label: 3.8.an_dc_alc
• 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:	$( 1 - 5 x + 8 x^{2} )( 1 - 4 x + 8 x^{2} )^{2}$
	$1 - 13 x + 80 x^{2} - 288 x^{3} + 640 x^{4} - 832 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.154919815756$, $\pm0.250000000000$, $\pm0.250000000000$
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:	$1$
Slopes:	$[0, 1/2, 1/2, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$100$	$236600$	$150888700$	$73972990000$	$36039459252500$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$56$	$572$	$4400$	$33556$	$263144$	$2095852$	$16766816$	$134192516$	$1073731736$

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

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

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

• 1.4096.bv : \(\Q(\sqrt{-7}) \).
• 1.4096.ey^2 : $\mathrm{M}_{2}(B)$, where $B$ is 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 $\times$ 1.64.a^2. The endomorphism algebra for each factor is:

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

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.ab_c_ag
$\F_{2}$	3.2.f_o_y

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.8.af_i_a	$2$	(not in LMFDB)
3.8.ad_a_bg	$2$	(not in LMFDB)
3.8.d_a_abg	$2$	(not in LMFDB)
3.8.f_i_a	$2$	(not in LMFDB)
3.8.n_dc_lc	$2$	(not in LMFDB)
3.8.ab_ae_y	$3$	(not in LMFDB)