Properties

Label 3.3.ae_e_a
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:  $( 1 - 3 x + 3 x^{2} )( 1 - x - 2 x^{2} - 3 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0734519173280$, $\pm0.166666666667$, $\pm0.740118583995$
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
$A(\F_{q^r})$ 4 336 11200 646464 14147284 406425600 11275287244 281074789632 7732512961600 206706589004496

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 2 12 98 240 764 2352 6530 19956 59282

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 2.3.ab_ac 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.ak 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.ac_ac_m$2$3.9.ai_bo_afi
3.3.c_ac_am$2$3.9.ai_bo_afi
3.3.e_e_a$2$3.9.ai_bo_afi
3.3.ab_b_ag$3$(not in LMFDB)
3.3.ab_e_aj$3$(not in LMFDB)
3.3.c_ac_am$3$(not in LMFDB)
3.3.c_k_m$3$(not in LMFDB)
3.3.f_q_bh$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ac_ac_m$2$3.9.ai_bo_afi
3.3.c_ac_am$2$3.9.ai_bo_afi
3.3.e_e_a$2$3.9.ai_bo_afi
3.3.ab_b_ag$3$(not in LMFDB)
3.3.ab_e_aj$3$(not in LMFDB)
3.3.c_ac_am$3$(not in LMFDB)
3.3.c_k_m$3$(not in LMFDB)
3.3.f_q_bh$3$(not in LMFDB)
3.3.af_q_abh$6$(not in LMFDB)
3.3.ad_i_ap$6$(not in LMFDB)
3.3.ac_k_am$6$(not in LMFDB)
3.3.a_i_a$6$(not in LMFDB)
3.3.b_b_g$6$(not in LMFDB)
3.3.b_e_j$6$(not in LMFDB)
3.3.d_i_p$6$(not in LMFDB)
3.3.ad_ac_p$12$(not in LMFDB)
3.3.a_ac_a$12$(not in LMFDB)
3.3.d_ac_ap$12$(not in LMFDB)