Abelian variety isogeny class 3.4.ah_x_aca over $\F_{2^{2}}$

• Label: 3.4.ah_x_aca
• Base field: $\F_{2^{2}}$
• 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:	$\F_{2^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x )^{2}( 1 - 3 x + 7 x^{2} - 12 x^{3} + 16 x^{4} )$
	$1 - 7 x + 23 x^{2} - 52 x^{3} + 92 x^{4} - 112 x^{5} + 64 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.190783854037$, $\pm0.524117187371$
Angle rank:	$1$ (numerical)

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9$	$3159$	$206388$	$14294475$	$1096788339$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$14$	$49$	$218$	$1048$	$4199$	$16126$	$64562$	$261121$	$1046054$

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 $\F_{2^{2}}$

The isogeny class factors as 1.4.ae $\times$ 2.4.ad_h and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.4.ae : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.4.ad_h : \(\Q(\sqrt{-3}, \sqrt{13})\).

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

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

• 1.4096.aey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4096.el^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 2.16.f_j. The endomorphism algebra for each factor is:

  • 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.16.f_j : \(\Q(\sqrt{-3}, \sqrt{13})\).
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq $\times$ 2.64.a_el. The endomorphism algebra for each factor is:

  • 1.64.aq : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.64.a_el : \(\Q(\sqrt{-3}, \sqrt{13})\).

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
3.4.ab_ab_ae	$2$	3.16.ad_ap_dk
3.4.b_ab_e	$2$	3.16.ad_ap_dk
3.4.h_x_ca	$2$	3.16.ad_ap_dk
3.4.ae_ab_u	$3$	(not in LMFDB)
3.4.ab_f_ak	$3$	(not in LMFDB)
3.4.c_ab_ak	$3$	(not in LMFDB)
3.4.f_r_bm	$3$	(not in LMFDB)