# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $4$ L-polynomial: $( 1 - 2 x + 3 x^{2} )^{4}$ Frobenius angles: $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$ 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 16 20736 2085136 84934656 3429742096 218889236736 19522293907216 1816019815366656 156005942621967376 12560354365595832576

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 18 68 138 236 546 1844 6426 20444 60978

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ac 4 and its endomorphism algebra is $\mathrm{M}_{4}($$$\Q(\sqrt{-2})$$$)$
All geometric endomorphisms are defined over $\F_{3}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ae_m_au_bm $2$ (not in LMFDB) 4.3.a_e_a_w $2$ (not in LMFDB) 4.3.e_m_u_bm $2$ (not in LMFDB) 4.3.i_bk_ea_ig $2$ (not in LMFDB) 4.3.ac_d_k_au $3$ (not in LMFDB) 4.3.e_g_q_br $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ae_m_au_bm $2$ (not in LMFDB) 4.3.a_e_a_w $2$ (not in LMFDB) 4.3.e_m_u_bm $2$ (not in LMFDB) 4.3.i_bk_ea_ig $2$ (not in LMFDB) 4.3.ac_d_k_au $3$ (not in LMFDB) 4.3.e_g_q_br $3$ (not in LMFDB) 4.3.ae_i_ae_ac $4$ (not in LMFDB) 4.3.a_ae_a_w $4$ (not in LMFDB) 4.3.a_a_a_o $4$ (not in LMFDB) 4.3.e_i_e_ac $4$ (not in LMFDB) 4.3.c_b_ae_al $5$ (not in LMFDB) 4.3.ag_t_abq_cy $6$ (not in LMFDB) 4.3.ae_g_aq_br $6$ (not in LMFDB) 4.3.ac_d_ak_u $6$ (not in LMFDB) 4.3.a_ac_a_af $6$ (not in LMFDB) 4.3.c_d_ak_au $6$ (not in LMFDB) 4.3.c_d_k_u $6$ (not in LMFDB) 4.3.g_t_bq_cy $6$ (not in LMFDB) 4.3.ai_bg_adk_gw $8$ (not in LMFDB) 4.3.ai_bi_ads_hm $8$ (not in LMFDB) 4.3.ae_g_ae_c $8$ (not in LMFDB) 4.3.ae_k_au_bi $8$ (not in LMFDB) 4.3.a_a_a_ao $8$ (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.e_g_e_c $8$ (not in LMFDB) 4.3.e_k_u_bi $8$ (not in LMFDB) 4.3.i_bg_dk_gw $8$ (not in LMFDB) 4.3.i_bi_ds_hm $8$ (not in LMFDB) 4.3.ac_b_e_al $10$ (not in LMFDB) 4.3.ac_ab_ac_q $12$ (not in LMFDB) 4.3.a_c_a_af $12$ (not in LMFDB) 4.3.c_ab_c_q $12$ (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.ae_i_ai_h $24$ (not in LMFDB) 4.3.ac_b_g_aw $24$ (not in LMFDB) 4.3.c_b_ag_aw $24$ (not in LMFDB) 4.3.e_i_i_h $24$ (not in LMFDB) 4.3.g_r_bm_cw $24$ (not in LMFDB)