Properties

Label 4.3.ai_bi_adr_hk
Base Field $\F_{3}$
Dimension $4$
Ordinary No
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 7 x^{2} - 9 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.227267020856$, $\pm0.304086723985$, $\pm0.464830336654$
Angle rank:  $3$ (numerical)

This isogeny class is not simple.

Newton polygon

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

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
$A(\F_{q^r})$ 10 12180 1122520 57002400 4066084000 311728294080 23231717873290 1799908010409600 147649492269375880 12186375001164672000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 14 47 106 281 803 2222 6370 19361 59189

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.ac $\times$ 2.3.ad_h 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.cc $\times$ 2.729.cn_dov. 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
4.3.ae_k_at_bk$2$(not in LMFDB)
4.3.ac_e_af_m$2$(not in LMFDB)
4.3.ac_e_b_a$2$(not in LMFDB)
4.3.c_e_ab_a$2$(not in LMFDB)
4.3.c_e_f_m$2$(not in LMFDB)
4.3.e_k_t_bk$2$(not in LMFDB)
4.3.i_bi_dr_hk$2$(not in LMFDB)
4.3.af_t_abv_ds$3$(not in LMFDB)
4.3.ac_e_b_a$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ae_k_at_bk$2$(not in LMFDB)
4.3.ac_e_af_m$2$(not in LMFDB)
4.3.ac_e_b_a$2$(not in LMFDB)
4.3.c_e_ab_a$2$(not in LMFDB)
4.3.c_e_f_m$2$(not in LMFDB)
4.3.e_k_t_bk$2$(not in LMFDB)
4.3.i_bi_dr_hk$2$(not in LMFDB)
4.3.af_t_abv_ds$3$(not in LMFDB)
4.3.ac_e_b_a$3$(not in LMFDB)
4.3.ab_h_ah_y$6$(not in LMFDB)
4.3.b_h_h_y$6$(not in LMFDB)
4.3.f_t_bv_ds$6$(not in LMFDB)