Properties

Label 4.3.ai_bi_ads_hm
Base Field $\F_{3}$
Dimension $4$
Ordinary Yes
$p$-rank $4$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 2 x + 3 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0540867239847$, $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.445913276015$
Angle rank:  $1$ (numerical)

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $4$
Slopes:  $[0, 0, 0, 0, 1, 1, 1, 1]$

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
$A(\F_{q^r})$ 8 9792 903944 42614784 2886151048 248946177600 21641775795592 1816019815366656 150768372127835144 12357774548336619072

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 14 44 82 196 638 2068 6426 19772 60014

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac 2 $\times$ 2.3.ae_i 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 $\F_{3^{8}}$ is 1.6561.abi 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-2}) \)$)$
All geometric endomorphisms are defined over $\F_{3^{8}}$.
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.ae_k_au_bi$2$(not in LMFDB)
4.3.a_c_ai_c$2$(not in LMFDB)
4.3.a_c_i_c$2$(not in LMFDB)
4.3.e_k_u_bi$2$(not in LMFDB)
4.3.i_bi_ds_hm$2$(not in LMFDB)
4.3.ac_b_g_aw$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ae_k_au_bi$2$(not in LMFDB)
4.3.a_c_ai_c$2$(not in LMFDB)
4.3.a_c_i_c$2$(not in LMFDB)
4.3.e_k_u_bi$2$(not in LMFDB)
4.3.i_bi_ds_hm$2$(not in LMFDB)
4.3.ac_b_g_aw$3$(not in LMFDB)
4.3.ae_g_ae_c$4$(not in LMFDB)
4.3.e_g_e_c$4$(not in LMFDB)
4.3.ag_r_abm_cw$6$(not in LMFDB)
4.3.c_b_ag_aw$6$(not in LMFDB)
4.3.g_r_bm_cw$6$(not in LMFDB)
4.3.ai_bg_adk_gw$8$(not in LMFDB)
4.3.ai_bk_aea_ig$8$(not in LMFDB)
4.3.ae_i_ae_ac$8$(not in LMFDB)
4.3.ae_m_au_bm$8$(not in LMFDB)
4.3.a_ae_a_w$8$(not in LMFDB)
4.3.a_a_a_ao$8$(not in LMFDB)
4.3.a_a_a_o$8$(not in LMFDB)
4.3.a_e_a_w$8$(not in LMFDB)
4.3.e_i_e_ac$8$(not in LMFDB)
4.3.e_m_u_bm$8$(not in LMFDB)
4.3.i_bg_dk_gw$8$(not in LMFDB)
4.3.i_bk_ea_ig$8$(not in LMFDB)
4.3.a_ai_a_bg$16$(not in LMFDB)
4.3.a_i_a_bg$16$(not in LMFDB)
4.3.ag_t_abq_cy$24$(not in LMFDB)
4.3.ae_g_aq_br$24$(not in LMFDB)
4.3.ae_i_ai_h$24$(not in LMFDB)
4.3.ac_ab_ac_q$24$(not in LMFDB)
4.3.ac_d_ak_u$24$(not in LMFDB)
4.3.ac_d_k_au$24$(not in LMFDB)
4.3.a_ac_a_af$24$(not in LMFDB)
4.3.a_c_a_af$24$(not in LMFDB)
4.3.c_ab_c_q$24$(not in LMFDB)
4.3.c_d_ak_au$24$(not in LMFDB)
4.3.c_d_k_u$24$(not in LMFDB)
4.3.e_g_q_br$24$(not in LMFDB)
4.3.e_i_i_h$24$(not in LMFDB)
4.3.g_t_bq_cy$24$(not in LMFDB)
4.3.ac_b_e_al$40$(not in LMFDB)
4.3.c_b_ae_al$40$(not in LMFDB)