Properties

Label 5.3.al_cg_aho_sp_abjx
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:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 45 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.0714477711956$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.272071776080$, $\pm0.560185743604$
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
$A(\F_{q^r})$ 5 37975 12822320 4011109375 1172505761275 235283417286400 49582326048864320 12257075126212109375 3008478155738955147920 717620959298433295774375

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -7 5 23 93 323 827 2170 6613 20039 59025

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 3.3.af_n_az 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.al_cmn_aoxb. 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.ab_ac_e_d_ad$2$(not in LMFDB)
5.3.f_k_k_j_p$2$(not in LMFDB)
5.3.l_cg_ho_sp_bjx$2$(not in LMFDB)
5.3.ai_bi_adz_jm_asg$3$(not in LMFDB)
5.3.af_k_ak_j_ap$3$(not in LMFDB)
5.3.af_t_acd_ew_ajg$3$(not in LMFDB)
5.3.ac_e_ah_g_ag$3$(not in LMFDB)
5.3.b_ac_ae_d_d$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
5.3.ab_ac_e_d_ad$2$(not in LMFDB)
5.3.f_k_k_j_p$2$(not in LMFDB)
5.3.l_cg_ho_sp_bjx$2$(not in LMFDB)
5.3.ai_bi_adz_jm_asg$3$(not in LMFDB)
5.3.af_k_ak_j_ap$3$(not in LMFDB)
5.3.af_t_acd_ew_ajg$3$(not in LMFDB)
5.3.ac_e_ah_g_ag$3$(not in LMFDB)
5.3.b_ac_ae_d_d$3$(not in LMFDB)
5.3.af_q_abo_dj_agj$4$(not in LMFDB)
5.3.f_q_bo_dj_gj$4$(not in LMFDB)
5.3.ac_e_ah_g_ag$6$(not in LMFDB)
5.3.c_e_h_g_g$6$(not in LMFDB)
5.3.f_t_cd_ew_jg$6$(not in LMFDB)
5.3.i_bi_dz_jm_sg$6$(not in LMFDB)
5.3.af_h_f_abe_ci$12$(not in LMFDB)
5.3.f_h_af_abe_aci$12$(not in LMFDB)
5.3.af_n_az_bw_adm$24$(not in LMFDB)
5.3.f_n_z_bw_dm$24$(not in LMFDB)