Properties

Label 5.3.ak_bt_aer_ix_apd
Base field $\F_{3}$
Dimension $5$
$p$-rank $3$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{3}$
Dimension:  $5$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 6 x^{2} - 7 x^{3} + 18 x^{4} - 36 x^{5} + 27 x^{6} )$
  $1 - 10 x + 45 x^{2} - 121 x^{3} + 231 x^{4} - 393 x^{5} + 693 x^{6} - 1089 x^{7} + 1215 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:  $\pm0.0452398905210$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.239335307006$, $\pm0.691360448188$
Angle rank:  $3$ (numerical)

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $5$ $24255$ $8988560$ $5267337075$ $1082283492775$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-6$ $0$ $15$ $116$ $304$ $765$ $2332$ $6484$ $19140$ $58800$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_g_ah 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.acv_eli_afacr. The endomorphism algebra for each factor is:
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_d_f_j_abz$2$(not in LMFDB)
5.3.ac_ad_n_d_abn$2$(not in LMFDB)
5.3.c_ad_an_d_bn$2$(not in LMFDB)
5.3.e_d_af_j_bz$2$(not in LMFDB)
5.3.k_bt_er_ix_pd$2$(not in LMFDB)
5.3.ah_y_acg_eq_aio$3$(not in LMFDB)
5.3.ae_d_f_j_abz$3$(not in LMFDB)
5.3.ae_m_abf_cl_aek$3$(not in LMFDB)
5.3.ab_a_ae_g_ag$3$(not in LMFDB)
5.3.c_ad_an_d_bn$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
5.3.ae_d_f_j_abz$2$(not in LMFDB)
5.3.ac_ad_n_d_abn$2$(not in LMFDB)
5.3.c_ad_an_d_bn$2$(not in LMFDB)
5.3.e_d_af_j_bz$2$(not in LMFDB)
5.3.k_bt_er_ix_pd$2$(not in LMFDB)
5.3.ah_y_acg_eq_aio$3$(not in LMFDB)
5.3.ae_d_f_j_abz$3$(not in LMFDB)
5.3.ae_m_abf_cl_aek$3$(not in LMFDB)
5.3.ab_a_ae_g_ag$3$(not in LMFDB)
5.3.c_ad_an_d_bn$3$(not in LMFDB)
5.3.ae_j_at_bt_adp$4$(not in LMFDB)
5.3.e_j_t_bt_dp$4$(not in LMFDB)
5.3.ah_y_acg_eq_aio$6$(not in LMFDB)
5.3.ae_d_f_j_abz$6$(not in LMFDB)
5.3.ae_m_abf_cl_aek$6$(not in LMFDB)
5.3.ac_ad_n_d_abn$6$(not in LMFDB)
5.3.ab_a_ae_g_ag$6$(not in LMFDB)
5.3.b_a_e_g_g$6$(not in LMFDB)
5.3.e_d_af_j_bz$6$(not in LMFDB)
5.3.e_m_bf_cl_ek$6$(not in LMFDB)
5.3.h_y_cg_eq_io$6$(not in LMFDB)
5.3.ae_a_r_aj_abe$12$(not in LMFDB)
5.3.ae_j_at_bt_adp$12$(not in LMFDB)
5.3.e_a_ar_aj_be$12$(not in LMFDB)
5.3.e_j_t_bt_dp$12$(not in LMFDB)
5.3.ae_g_ah_bb_acu$24$(not in LMFDB)
5.3.e_g_h_bb_cu$24$(not in LMFDB)