Abelian Variety Isogeny Class 4.2.ad_c_a_b over $\F_{2}$

• Label: 4.2.ad_c_a_b
• Base Field: $\F_{2}$
• Dimension: $4$
• Ordinary: No
• $p$-rank: $4$
• Principally polarizable: Yes

Invariants

Base field:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$1 - 3 x + 2 x^{2} + x^{4} + 8 x^{6} - 24 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.0298810195513$, $\pm0.106143893905$, $\pm0.506143893905$, $\pm0.829881019551$
Angle rank:	$2$ (numerical)
Number field:	8.0.26265625.1
Galois group:	$C_2^2:C_4$

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Point counts of the abelian variety

$r$	1	2	3	4	5	6	7	8	9	10
$A(\F_{q^r})$	1	55	1621	27775	609961	24161005	223259821	4066287775	72597199861	1153207515625

Point counts of the (virtual) curve

$r$	1	2	3	4	5	6	7	8	9	10
$C(\F_{q^r})$	0	0	0	4	15	90	105	244	540	1075

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 8.0.26265625.1.

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

The base change of $A$ to $\F_{2^{5}}$ is 2.32.aj_cb^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.1025.1$)$

All geometric endomorphisms are defined over $\F_{2^{5}}$.

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
4.2.d_c_a_b	$2$	4.4.af_g_u_adb
4.2.c_c_f_l	$5$	(not in LMFDB)
4.2.c_h_k_v	$5$	(not in LMFDB)

Additional information

This is the isogeny class of the Jacobian of a function field of class number 1. This example was found by Stirpe in 2014 [10.1016/j.jnt.2014.02.016, MR:3227356], refuting a claim made in 1975 by Leitzel-Madan-Queen [0.1016/0022-314X(75)90004-9, MR:0369326].