Properties

Label 3.2.af_n_aw
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

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Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$
Frobenius angles:  $\pm0.123548644961$, $\pm0.250000000000$, $\pm0.456881978294$
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
$A(\F_{q^r})$ 1 95 988 4275 39401 375440 2491763 15736275 125716084 1141643975

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 6 13 18 38 87 152 242 481 1086

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac $\times$ 2.2.ad_f 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 $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.ey. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{12}}$.
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.ab_b_ac$2$3.4.b_b_i
3.2.b_b_c$2$3.4.b_b_i
3.2.f_n_w$2$3.4.b_b_i
3.2.ac_b_c$3$3.8.e_t_bs
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.ab_b_ac$2$3.4.b_b_i
3.2.b_b_c$2$3.4.b_b_i
3.2.f_n_w$2$3.4.b_b_i
3.2.ac_b_c$3$3.8.e_t_bs
3.2.f_n_w$4$3.16.b_ah_bo
3.2.c_b_ac$6$(not in LMFDB)
3.2.ad_h_am$8$(not in LMFDB)
3.2.d_h_m$8$(not in LMFDB)
3.2.ac_d_ac$12$(not in LMFDB)
3.2.c_d_c$12$(not in LMFDB)
3.2.a_b_a$24$(not in LMFDB)
3.2.a_d_a$24$(not in LMFDB)