Properties

Label 5.3.ak_by_agk_pv_abes
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:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 11 x^{2} - 22 x^{3} + 33 x^{4} - 36 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.132091637252$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.376445424065$, $\pm0.544359499442$
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
$A(\F_{q^r})$ 10 65660 17271520 3772823600 1079586372050 252212275911680 55134360098537590 12944967590790726400 3045381532607929772320 711872319466612931318300

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -6 10 30 86 304 880 2402 6974 20280 58550

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_l_aw 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.bq_bex_fga. 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_i_ak_j_ag$2$(not in LMFDB)
5.3.ac_c_ac_d_g$2$(not in LMFDB)
5.3.c_c_c_d_ag$2$(not in LMFDB)
5.3.e_i_k_j_g$2$(not in LMFDB)
5.3.k_by_gk_pv_bes$2$(not in LMFDB)
5.3.ah_bd_adk_ic_apm$3$(not in LMFDB)
5.3.ae_i_ak_j_ag$3$(not in LMFDB)
5.3.ae_r_abu_ee_ahw$3$(not in LMFDB)
5.3.ab_f_ae_g_ag$3$(not in LMFDB)
5.3.c_c_c_d_ag$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
5.3.ae_i_ak_j_ag$2$(not in LMFDB)
5.3.ac_c_ac_d_g$2$(not in LMFDB)
5.3.c_c_c_d_ag$2$(not in LMFDB)
5.3.e_i_k_j_g$2$(not in LMFDB)
5.3.k_by_gk_pv_bes$2$(not in LMFDB)
5.3.ah_bd_adk_ic_apm$3$(not in LMFDB)
5.3.ae_i_ak_j_ag$3$(not in LMFDB)
5.3.ae_r_abu_ee_ahw$3$(not in LMFDB)
5.3.ab_f_ae_g_ag$3$(not in LMFDB)
5.3.c_c_c_d_ag$3$(not in LMFDB)
5.3.ae_o_abi_cx_afi$4$(not in LMFDB)
5.3.e_o_bi_cx_fi$4$(not in LMFDB)
5.3.b_f_e_g_g$6$(not in LMFDB)
5.3.e_r_bu_ee_hw$6$(not in LMFDB)
5.3.h_bd_dk_ic_pm$6$(not in LMFDB)
5.3.ae_f_c_ay_ci$12$(not in LMFDB)
5.3.e_f_ac_ay_aci$12$(not in LMFDB)
5.3.ae_l_aw_bq_acu$24$(not in LMFDB)
5.3.e_l_w_bq_cu$24$(not in LMFDB)