Properties

Label 3.2.ae_l_as
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

Learn more about

Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )^{2}$
Frobenius angles:  $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.384973271919$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4 320 2548 6400 19844 203840 2278532 18662400 129063844 960449600

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 11 23 23 19 47 139 287 491 911

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac $\times$ 1.2.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:
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab 2 $\times$ 1.16.i. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.ac_f_ag$2$3.4.g_v_bw
3.2.a_d_ac$2$3.4.g_v_bw
3.2.a_d_c$2$3.4.g_v_bw
3.2.c_f_g$2$3.4.g_v_bw
3.2.e_l_s$2$3.4.g_v_bw
3.2.ab_ab_g$3$3.8.o_dl_mm
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.ac_f_ag$2$3.4.g_v_bw
3.2.a_d_ac$2$3.4.g_v_bw
3.2.a_d_c$2$3.4.g_v_bw
3.2.c_f_g$2$3.4.g_v_bw
3.2.e_l_s$2$3.4.g_v_bw
3.2.ab_ab_g$3$3.8.o_dl_mm
3.2.ac_ab_g$4$3.16.g_bh_hs
3.2.c_ab_ag$4$3.16.g_bh_hs
3.2.e_l_s$4$3.16.g_bh_hs
3.2.ad_d_ac$6$(not in LMFDB)
3.2.b_ab_ag$6$(not in LMFDB)
3.2.d_d_c$6$(not in LMFDB)
3.2.ac_h_ai$8$(not in LMFDB)
3.2.a_ab_a$8$(not in LMFDB)
3.2.a_f_a$8$(not in LMFDB)
3.2.c_h_i$8$(not in LMFDB)
3.2.ab_b_ae$24$(not in LMFDB)
3.2.b_b_e$24$(not in LMFDB)