Properties

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

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 + x + 3 x^{2} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.593214749339$
Angle rank:  $2$ (numerical)
Jacobians:  1

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 contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 10 1260 21280 655200 18035050 386104320 10039666990 289464739200 7753580353120 205065119328300

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 14 30 98 300 728 2100 6722 20010 58814

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.ac $\times$ 1.3.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 $\times$ 1.729.ak $\times$ 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.ag_u_abq$2$3.9.e_q_bq
3.3.ac_e_ag$2$3.9.e_q_bq
3.3.a_c_ag$2$3.9.e_q_bq
3.3.a_c_g$2$3.9.e_q_bq
3.3.c_e_g$2$3.9.e_q_bq
3.3.e_k_s$2$3.9.e_q_bq
3.3.g_u_bq$2$3.9.e_q_bq
3.3.ab_h_ag$3$(not in LMFDB)
3.3.c_e_g$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ag_u_abq$2$3.9.e_q_bq
3.3.ac_e_ag$2$3.9.e_q_bq
3.3.a_c_ag$2$3.9.e_q_bq
3.3.a_c_g$2$3.9.e_q_bq
3.3.c_e_g$2$3.9.e_q_bq
3.3.e_k_s$2$3.9.e_q_bq
3.3.g_u_bq$2$3.9.e_q_bq
3.3.ab_h_ag$3$(not in LMFDB)
3.3.c_e_g$3$(not in LMFDB)
3.3.ad_l_as$6$(not in LMFDB)
3.3.b_h_g$6$(not in LMFDB)
3.3.d_l_s$6$(not in LMFDB)