Abelian variety isogeny class 3.2.ad_g_ak over $\F_{2}$

• Label: 3.2.ad_g_ak
• Base field: $\F_{2}$
• Dimension: $3$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$3$
L-polynomial:	$( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )$
	$1 - 3 x + 6 x^{2} - 10 x^{3} + 12 x^{4} - 12 x^{5} + 8 x^{6}$
Frobenius angles:	$\pm0.0833333333333$, $\pm0.384973271919$, $\pm0.583333333333$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Isomorphism classes:	4

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$104$	$350$	$2704$	$29062$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$8$	$6$	$8$	$30$	$56$	$126$	$320$	$582$	$968$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2+(x^4+x^2+1)y=x^8+x^5+x^4+x^2+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ab $\times$ 2.2.ac_c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.ab : \(\Q(\sqrt{-7}) \).
• 2.2.ac_c : \(\Q(\zeta_{12})\).

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

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 $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.d $\times$ 2.4.a_ae. The endomorphism algebra for each factor is:

  • 1.4.d : \(\Q(\sqrt{-7}) \).
  • 2.4.a_ae : \(\Q(\zeta_{12})\).
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.ae^2 $\times$ 1.8.f. The endomorphism algebra for each factor is:

  • 1.8.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.8.f : \(\Q(\sqrt{-7}) \).
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae^2 $\times$ 1.16.ab. The endomorphism algebra for each factor is:

  • 1.16.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.16.ab : \(\Q(\sqrt{-7}) \).
• 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 is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.2.ab_c_ag	$2$	3.4.d_a_am
3.2.b_c_g	$2$	3.4.d_a_am
3.2.d_g_k	$2$	3.4.d_a_am
3.2.d_g_i	$3$	3.8.ad_a_bg