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

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

Invariants

Base field:	$\F_{2}$
Dimension:	$3$
L-polynomial:	$( 1 + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$
	$1 - 3 x + 7 x^{2} - 12 x^{3} + 14 x^{4} - 12 x^{5} + 8 x^{6}$
Frobenius angles:	$\pm0.123548644961$, $\pm0.456881978294$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Isomorphism classes:	1

This isogeny class is not simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3$	$171$	$684$	$1539$	$31713$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$10$	$9$	$2$	$30$	$103$	$168$	$242$	$513$	$1150$

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 $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.a $\times$ 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.a : \(\Q(\sqrt{-2}) \).
• 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.l^2 $\times$ 1.64.q. The endomorphism algebra for each factor is:

• 1.64.l^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
• 1.64.q : 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.e $\times$ 2.4.b_ad. The endomorphism algebra for each factor is:

  • 1.4.e : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 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.a $\times$ 2.8.a_l. The endomorphism algebra for each factor is:

  • 1.8.a : \(\Q(\sqrt{-2}) \).
  • 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.d_h_m	$2$	3.4.f_f_ae
3.2.a_b_a	$3$	3.8.a_t_a
3.2.af_n_aw	$8$	(not in LMFDB)