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

• Label: 3.4.ag_v_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:	$( 1 - 2 x )^{2}( 1 - x + 4 x^{2} )^{2}$
	$1 - 6 x + 21 x^{2} - 52 x^{3} + 84 x^{4} - 96 x^{5} + 64 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.419569376745$, $\pm0.419569376745$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$5184$	$283024$	$12960000$	$893053456$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$23$	$71$	$191$	$839$	$3983$	$16631$	$65471$	$259559$	$1043183$

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

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

The isogeny class factors as 1.4.ae $\times$ 1.4.ab^2 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 $\infty$.
• 1.4.ab^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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.ae_l_abc	$2$	3.16.g_ap_ahs
3.4.ac_f_au	$2$	3.16.g_ap_ahs
3.4.c_f_u	$2$	3.16.g_ap_ahs
3.4.e_l_bc	$2$	3.16.g_ap_ahs
3.4.g_v_ca	$2$	3.16.g_ap_ahs
3.4.ad_ad_u	$3$	(not in LMFDB)
3.4.a_j_c	$3$	(not in LMFDB)
3.4.d_d_c	$3$	(not in LMFDB)