Properties

Label 4.3.aj_bo_aek_ix
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:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 3 x + 7 x^{2} - 9 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.227267020856$, $\pm0.464830336654$
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
$A(\F_{q^r})$ 5 7105 827120 54033525 4553342000 357302606080 25077434754755 1831615948705725 146237929245768080 12037965374815072000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -5 9 37 101 310 903 2389 6485 19171 58464

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 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.cc 2 $\times$ 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.ad_e_ag_p$2$(not in LMFDB)
4.3.ad_e_a_ad$2$(not in LMFDB)
4.3.d_e_a_ad$2$(not in LMFDB)
4.3.d_e_g_p$2$(not in LMFDB)
4.3.j_bo_ek_ix$2$(not in LMFDB)
4.3.ag_w_acf_ek$3$(not in LMFDB)
4.3.ad_n_abb_ci$3$(not in LMFDB)
4.3.a_e_d_g$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ad_e_ag_p$2$(not in LMFDB)
4.3.ad_e_a_ad$2$(not in LMFDB)
4.3.d_e_a_ad$2$(not in LMFDB)
4.3.d_e_g_p$2$(not in LMFDB)
4.3.j_bo_ek_ix$2$(not in LMFDB)
4.3.ag_w_acf_ek$3$(not in LMFDB)
4.3.ad_n_abb_ci$3$(not in LMFDB)
4.3.a_e_d_g$3$(not in LMFDB)
4.3.ad_k_as_bn$4$(not in LMFDB)
4.3.d_k_s_bn$4$(not in LMFDB)
4.3.a_e_ad_g$6$(not in LMFDB)
4.3.d_n_bb_ci$6$(not in LMFDB)
4.3.g_w_cf_ek$6$(not in LMFDB)
4.3.ad_b_j_ay$12$(not in LMFDB)
4.3.d_b_aj_ay$12$(not in LMFDB)
4.3.ad_h_aj_s$24$(not in LMFDB)
4.3.d_h_j_s$24$(not in LMFDB)