Properties

Label 3.2.ae_k_ar
Base field $\F_{2}$
Dimension $3$
$p$-rank $3$
Ordinary yes
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:  $( 1 - x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$
  $1 - 4 x + 10 x^{2} - 17 x^{3} + 20 x^{4} - 16 x^{5} + 8 x^{6}$
Frobenius angles:  $\pm0.123548644961$, $\pm0.384973271919$, $\pm0.456881978294$
Angle rank:  $2$ (numerical)
Jacobians:  $0$
Isomorphism classes:  1

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2$ $152$ $1064$ $2736$ $21142$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $9$ $14$ $9$ $19$ $78$ $181$ $305$ $518$ $1029$

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 $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ab $\times$ 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 $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.l 2 . The endomorphism algebra for each factor is:
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_e_ah$2$3.4.e_e_ab
3.2.c_e_h$2$3.4.e_e_ab
3.2.e_k_r$2$3.4.e_e_ab
3.2.ab_b_b$3$3.8.f_t_cd

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
3.2.ac_e_ah$2$3.4.e_e_ab
3.2.c_e_h$2$3.4.e_e_ab
3.2.e_k_r$2$3.4.e_e_ab
3.2.ab_b_b$3$3.8.f_t_cd
3.2.b_b_ab$6$(not in LMFDB)
3.2.ab_d_ab$12$(not in LMFDB)
3.2.b_d_b$12$(not in LMFDB)