Properties

Label 5.3.ak_bx_agb_ol_abbp
Base Field $\F_{3}$
Dimension $5$
Ordinary No
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $5$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 10 x^{2} - 19 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.125412673718$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.335294135736$, $\pm0.584823404300$
Angle rank:  $3$ (numerical)

This isogeny class is not simple.

Newton polygon

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

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
$A(\F_{q^r})$ 9 56007 14881104 4514892291 1174205479779 232624036802304 52709453025633189 13118037723941971275 3065690147851241616384 713523065769308099282607

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -6 8 27 100 324 821 2304 7060 20412 58688

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_k_at 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 $\times$ 3.729.ar_fe_abgaf. 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
5.3.ae_h_ah_j_ap$2$(not in LMFDB)
5.3.ac_b_b_d_ad$2$(not in LMFDB)
5.3.c_b_ab_d_d$2$(not in LMFDB)
5.3.e_h_h_j_p$2$(not in LMFDB)
5.3.k_bx_gb_ol_bbp$2$(not in LMFDB)
5.3.ah_bc_ade_hk_aoc$3$(not in LMFDB)
5.3.ae_h_ah_j_ap$3$(not in LMFDB)
5.3.ae_q_abr_dv_ahe$3$(not in LMFDB)
5.3.ab_e_ae_g_ag$3$(not in LMFDB)
5.3.c_b_ab_d_d$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
5.3.ae_h_ah_j_ap$2$(not in LMFDB)
5.3.ac_b_b_d_ad$2$(not in LMFDB)
5.3.c_b_ab_d_d$2$(not in LMFDB)
5.3.e_h_h_j_p$2$(not in LMFDB)
5.3.k_bx_gb_ol_bbp$2$(not in LMFDB)
5.3.ah_bc_ade_hk_aoc$3$(not in LMFDB)
5.3.ae_h_ah_j_ap$3$(not in LMFDB)
5.3.ae_q_abr_dv_ahe$3$(not in LMFDB)
5.3.ab_e_ae_g_ag$3$(not in LMFDB)
5.3.c_b_ab_d_d$3$(not in LMFDB)
5.3.ae_n_abf_cr_aez$4$(not in LMFDB)
5.3.e_n_bf_cr_ez$4$(not in LMFDB)
5.3.ah_bc_ade_hk_aoc$6$(not in LMFDB)
5.3.ae_h_ah_j_ap$6$(not in LMFDB)
5.3.ae_q_abr_dv_ahe$6$(not in LMFDB)
5.3.ac_b_b_d_ad$6$(not in LMFDB)
5.3.ab_e_ae_g_ag$6$(not in LMFDB)
5.3.b_e_e_g_g$6$(not in LMFDB)
5.3.e_h_h_j_p$6$(not in LMFDB)
5.3.e_q_br_dv_he$6$(not in LMFDB)
5.3.h_bc_de_hk_oc$6$(not in LMFDB)
5.3.ae_e_f_av_bq$12$(not in LMFDB)
5.3.ae_n_abf_cr_aez$12$(not in LMFDB)
5.3.e_e_af_av_abq$12$(not in LMFDB)
5.3.e_n_bf_cr_ez$12$(not in LMFDB)
5.3.ae_k_at_bn_acu$24$(not in LMFDB)
5.3.e_k_t_bn_cu$24$(not in LMFDB)