Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{2}$
Frobenius angles:  $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.5$
Angle rank:  $1$ (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
$A(\F_{q^r})$ 16 2304 40432 589824 14289616 366799104 9667467952 272734617600 7774705143184 210999098052864

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 20 48 92 240 692 2016 6332 20064 60500

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac 2 $\times$ 1.3.a 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 $\F_{3^{2}}$ is 1.9.c 2 $\times$ 1.9.g. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{2}}$.

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.a_f_a$2$3.9.k_cd_hw
3.3.e_n_y$2$3.9.k_cd_hw
3.3.ah_z_acc$3$(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.a_f_a$2$3.9.k_cd_hw
3.3.e_n_y$2$3.9.k_cd_hw
3.3.ah_z_acc$3$(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.a_b_a$4$(not in LMFDB)
3.3.af_k_ap$6$(not in LMFDB)
3.3.ad_f_ag$6$(not in LMFDB)
3.3.ac_e_am$6$(not in LMFDB)
3.3.ab_b_g$6$(not in LMFDB)
3.3.b_ac_aj$6$(not in LMFDB)
3.3.b_b_ag$6$(not in LMFDB)
3.3.d_f_g$6$(not in LMFDB)
3.3.h_z_cc$6$(not in LMFDB)
3.3.ae_l_ay$8$(not in LMFDB)
3.3.e_l_y$8$(not in LMFDB)
3.3.ad_b_g$12$(not in LMFDB)
3.3.d_b_ag$12$(not in LMFDB)
3.3.ah_x_abw$24$(not in LMFDB)
3.3.ab_ab_a$24$(not in LMFDB)
3.3.b_ab_a$24$(not in LMFDB)
3.3.h_x_bw$24$(not in LMFDB)