# 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

## 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 8 9792 903944 42614784 2886151048 248946177600 21641775795592 1816019815366656 150768372127835144 12357774548336619072

 $r$ 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: 1.3.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 2.3.ae_i : $$\Q(\zeta_{8})$$.
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
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.c 2 $\times$ 2.9.a_ao. The endomorphism algebra for each factor is: 1.9.c 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 2.9.a_ao : $$\Q(\zeta_{8})$$.
• Endomorphism algebra over $\F_{3^{4}}$  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 2 $\times$ 1.81.o 2 . The endomorphism algebra for each factor is: 1.81.ao 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.81.o 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$

## 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_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.
 Twist Extension Degree Common 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)