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

• Label: 3.4.ai_bc_acm
• Base field: $\F_{2^{2}}$
• Dimension: $3$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x )^{4}( 1 + 4 x^{2} )$
	$1 - 8 x + 28 x^{2} - 64 x^{3} + 112 x^{4} - 128 x^{5} + 64 x^{6}$
Frobenius angles:	$0$, $0$, $0$, $0$, $\pm0.5$
Angle rank:	$0$ (numerical)

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5$	$2025$	$156065$	$11390625$	$946609025$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$9$	$33$	$161$	$897$	$3969$	$15873$	$64001$	$260097$	$1046529$

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 $\F_{2^{8}}$.

Endomorphism algebra over $\F_{2^{2}}$

The isogeny class factors as 1.4.ae^2 $\times$ 1.4.a 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^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4.a : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

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

  • 1.16.ai^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

Subfield	Primitive Model
$\F_{2}$	3.2.ac_ac_i
$\F_{2}$	3.2.c_ac_ai

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.4.a_ae_a	$2$	3.16.ai_aq_jw
3.4.i_bc_cm	$2$	3.16.ai_aq_jw
3.4.ac_e_aq	$3$	(not in LMFDB)
3.4.e_q_bg	$3$	(not in LMFDB)