Properties

Label 4.3.ah_z_acl_et
Base Field $\F_{3}$
Dimension $4$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 10 x^{2} - 21 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.0145064862012$, $\pm0.166666666667$, $\pm0.383559653096$, $\pm0.564732805964$
Angle rank:  $2$ (numerical)

This isogeny class is not simple.

Newton polygon

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

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
$A(\F_{q^r})$ 7 6321 407092 32622681 3349212307 277908681456 22350531758359 1880571277251849 150525368480881408 11889101913430315161

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 11 21 59 237 719 2139 6659 19740 57731

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 3.3.ae_k_av 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.cc $\times$ 3.729.acn_djq_adrer. 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.ab_b_ad_ad$2$(not in LMFDB)
4.3.b_b_d_ad$2$(not in LMFDB)
4.3.h_z_cl_et$2$(not in LMFDB)
4.3.ae_n_abh_ci$3$(not in LMFDB)
4.3.ab_af_d_p$3$(not in LMFDB)
4.3.ab_b_ad_ad$3$(not in LMFDB)
4.3.ab_b_ad_p$3$(not in LMFDB)
4.3.c_b_ad_am$3$(not in LMFDB)
4.3.c_h_j_y$3$(not in LMFDB)
4.3.f_h_aj_abn$3$(not in LMFDB)
4.3.f_n_v_bh$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ab_b_ad_ad$2$(not in LMFDB)
4.3.b_b_d_ad$2$(not in LMFDB)
4.3.h_z_cl_et$2$(not in LMFDB)
4.3.ae_n_abh_ci$3$(not in LMFDB)
4.3.ab_af_d_p$3$(not in LMFDB)
4.3.ab_b_ad_ad$3$(not in LMFDB)
4.3.ab_b_ad_p$3$(not in LMFDB)
4.3.c_b_ad_am$3$(not in LMFDB)
4.3.c_h_j_y$3$(not in LMFDB)
4.3.f_h_aj_abn$3$(not in LMFDB)
4.3.f_n_v_bh$3$(not in LMFDB)
4.3.af_h_j_abn$6$(not in LMFDB)
4.3.af_n_av_bh$6$(not in LMFDB)
4.3.ac_b_d_am$6$(not in LMFDB)
4.3.ac_h_aj_y$6$(not in LMFDB)
4.3.b_af_ad_p$6$(not in LMFDB)
4.3.b_b_d_p$6$(not in LMFDB)
4.3.e_n_bh_ci$6$(not in LMFDB)