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

• Label: 3.2.ac_e_ah
• Base field: $\F_{2}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• 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:	$( 1 + x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$
	$1 - 2 x + 4 x^{2} - 7 x^{3} + 8 x^{4} - 8 x^{5} + 8 x^{6}$
Frobenius angles:	$\pm0.123548644961$, $\pm0.456881978294$, $\pm0.615026728081$
Angle rank:	$2$ (numerical)
Jacobians:	$1$
Isomorphism classes:	3

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

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$152$	$304$	$2736$	$42284$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$9$	$4$	$9$	$41$	$78$	$155$	$305$	$508$	$1029$

Jacobians and polarizations

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

• $y^2+(x^4+x^2+x+1)y=x^8+x^5+x^4+x^3+x+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.b $\times$ 2.2.ad_f 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.b : \(\Q(\sqrt{-7}) \).
• 2.2.ad_f : \(\Q(\sqrt{-3}, \sqrt{5})\).

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

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

• 1.64.aj : \(\Q(\sqrt{-7}) \).
• 1.64.l^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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.b_ad. The endomorphism algebra for each factor is:

  • 1.4.d : \(\Q(\sqrt{-7}) \).
  • 2.4.b_ad : \(\Q(\sqrt{-3}, \sqrt{5})\).
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.af $\times$ 2.8.a_l. The endomorphism algebra for each factor is:

  • 1.8.af : \(\Q(\sqrt{-7}) \).
  • 2.8.a_l : \(\Q(\sqrt{-3}, \sqrt{5})\).

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.ae_k_ar	$2$	3.4.e_e_ab
3.2.c_e_h	$2$	3.4.e_e_ab
3.2.e_k_r	$2$	3.4.e_e_ab
3.2.b_b_ab	$3$	3.8.af_t_acd
3.2.e_k_r	$3$	3.8.af_t_acd