Properties

Label 3.3.af_k_ap
Base Field $\F_{3}$
Dimension $3$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0292466093486$, $\pm0.166666666667$, $\pm0.637420057318$
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
$A(\F_{q^r})$ 3 399 9072 503139 15805533 366799104 10437424959 287405575275 7480235652624 203383749396639

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 5 8 77 269 692 2183 6677 19304 58325

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 2.3.ac_b 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 $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu 2 $\times$ 1.729.cc. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{6}}$.
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
3.3.ab_ac_j$2$3.9.af_k_av
3.3.b_ac_aj$2$3.9.af_k_av
3.3.f_k_p$2$3.9.af_k_av
3.3.ac_e_am$3$(not in LMFDB)
3.3.b_ac_aj$3$(not in LMFDB)
3.3.b_b_ag$3$(not in LMFDB)
3.3.e_n_y$3$(not in LMFDB)
3.3.h_z_cc$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ab_ac_j$2$3.9.af_k_av
3.3.b_ac_aj$2$3.9.af_k_av
3.3.f_k_p$2$3.9.af_k_av
3.3.ac_e_am$3$(not in LMFDB)
3.3.b_ac_aj$3$(not in LMFDB)
3.3.b_b_ag$3$(not in LMFDB)
3.3.e_n_y$3$(not in LMFDB)
3.3.h_z_cc$3$(not in LMFDB)
3.3.ah_z_acc$6$(not in LMFDB)
3.3.ae_n_ay$6$(not in LMFDB)
3.3.ad_f_ag$6$(not in LMFDB)
3.3.ab_b_g$6$(not in LMFDB)
3.3.a_f_a$6$(not in LMFDB)
3.3.c_e_m$6$(not in LMFDB)
3.3.d_f_g$6$(not in LMFDB)
3.3.ad_b_g$12$(not in LMFDB)
3.3.a_b_a$12$(not in LMFDB)
3.3.d_b_ag$12$(not in LMFDB)
3.3.ah_x_abw$24$(not in LMFDB)
3.3.ae_l_ay$24$(not in LMFDB)
3.3.ab_ab_a$24$(not in LMFDB)
3.3.b_ab_a$24$(not in LMFDB)
3.3.e_l_y$24$(not in LMFDB)
3.3.h_x_bw$24$(not in LMFDB)