Properties

Label 3.3.ah_z_acc
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:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{2}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.304086723985$
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 1008 40432 838656 15870844 366799104 10025358676 283090010112 7774705143184 208415479024368

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 11 48 119 267 692 2097 6575 20064 59771

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.ac 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 $\overline{\F}_{3}$
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.ad_f_ag$2$3.9.b_t_g
3.3.ab_b_g$2$3.9.b_t_g
3.3.b_b_ag$2$3.9.b_t_g
3.3.d_f_g$2$3.9.b_t_g
3.3.h_z_cc$2$3.9.b_t_g
3.3.ae_n_ay$3$(not in LMFDB)
3.3.ab_ac_j$3$(not in LMFDB)
3.3.ab_b_g$3$(not in LMFDB)
3.3.c_e_m$3$(not in LMFDB)
3.3.f_k_p$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ad_f_ag$2$3.9.b_t_g
3.3.ab_b_g$2$3.9.b_t_g
3.3.b_b_ag$2$3.9.b_t_g
3.3.d_f_g$2$3.9.b_t_g
3.3.h_z_cc$2$3.9.b_t_g
3.3.ae_n_ay$3$(not in LMFDB)
3.3.ab_ac_j$3$(not in LMFDB)
3.3.ab_b_g$3$(not in LMFDB)
3.3.c_e_m$3$(not in LMFDB)
3.3.f_k_p$3$(not in LMFDB)
3.3.ad_b_g$4$(not in LMFDB)
3.3.d_b_ag$4$(not in LMFDB)
3.3.af_k_ap$6$(not in LMFDB)
3.3.ac_e_am$6$(not in LMFDB)
3.3.a_f_a$6$(not in LMFDB)
3.3.b_ac_aj$6$(not in LMFDB)
3.3.e_n_y$6$(not in LMFDB)
3.3.ah_x_abw$8$(not in LMFDB)
3.3.ab_ab_a$8$(not in LMFDB)
3.3.b_ab_a$8$(not in LMFDB)
3.3.h_x_bw$8$(not in LMFDB)
3.3.a_b_a$12$(not in LMFDB)
3.3.ae_l_ay$24$(not in LMFDB)
3.3.e_l_y$24$(not in LMFDB)