Properties

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

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )^{2}$
Frobenius angles:  $\pm0.0540867239847$, $\pm0.0540867239847$, $\pm0.445913276015$, $\pm0.445913276015$
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
$A(\F_{q^r})$ 4 4624 391876 21381376 2428715524 283130410000 23991364017604 1816019815366656 145706642016575236 12158462041948282384

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 10 20 26 156 730 2292 6426 19100 59050

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 2.3.ae_i 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-2}) \)$)$
All geometric endomorphisms are defined over $\F_{3^{4}}$.
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.a_a_a_ao$2$(not in LMFDB)
4.3.i_bg_dk_gw$2$(not in LMFDB)
4.3.e_i_i_h$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.a_a_a_ao$2$(not in LMFDB)
4.3.i_bg_dk_gw$2$(not in LMFDB)
4.3.e_i_i_h$3$(not in LMFDB)
4.3.ae_i_ai_h$6$(not in LMFDB)
4.3.ai_bi_ads_hm$8$(not in LMFDB)
4.3.ai_bk_aea_ig$8$(not in LMFDB)
4.3.ae_g_ae_c$8$(not in LMFDB)
4.3.ae_i_ae_ac$8$(not in LMFDB)
4.3.ae_k_au_bi$8$(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_o$8$(not in LMFDB)
4.3.a_c_ai_c$8$(not in LMFDB)
4.3.a_c_i_c$8$(not in LMFDB)
4.3.a_e_a_w$8$(not in LMFDB)
4.3.e_g_e_c$8$(not in LMFDB)
4.3.e_i_e_ac$8$(not in LMFDB)
4.3.e_k_u_bi$8$(not in LMFDB)
4.3.e_m_u_bm$8$(not in LMFDB)
4.3.i_bi_ds_hm$8$(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_r_abm_cw$24$(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.ac_ab_ac_q$24$(not in LMFDB)
4.3.ac_b_g_aw$24$(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_b_ag_aw$24$(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.g_r_bm_cw$24$(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)