# Properties

 Label 4.3.ak_bx_afu_ma Base Field $\F_{3}$ Dimension $4$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $4$ L-polynomial: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 3 x^{2} )^{2}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.304086723985$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $2$ Slopes: $[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$

## 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 $A(\F_{q^r})$ 4 7056 1132096 76317696 4300998724 287570497536 22747538835844 1880566937174016 153037296038433856 12256289074986008976

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 8 48 128 294 746 2178 6656 20064 59528

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 1.3.ac 2 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.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.3.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu 2 $\times$ 1.729.cc 2 . The endomorphism algebra for each factor is: 1.729.abu 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.729.cc 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad 2 $\times$ 1.9.c 2 . The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.9.c 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 1.27.k 2 . The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.27.k 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.ag_r_abe_bw $2$ (not in LMFDB) 4.3.ae_h_a_am $2$ (not in LMFDB) 4.3.ac_b_ag_y $2$ (not in LMFDB) 4.3.a_ab_a_m $2$ (not in LMFDB) 4.3.c_b_g_y $2$ (not in LMFDB) 4.3.e_h_a_am $2$ (not in LMFDB) 4.3.g_r_be_bw $2$ (not in LMFDB) 4.3.k_bx_fu_ma $2$ (not in LMFDB) 4.3.ah_bc_acx_fu $3$ (not in LMFDB) 4.3.ae_e_m_abn $3$ (not in LMFDB) 4.3.ae_q_abk_da $3$ (not in LMFDB) 4.3.ab_b_g_am $3$ (not in LMFDB) 4.3.ab_e_d_g $3$ (not in LMFDB) 4.3.c_ac_a_p $3$ (not in LMFDB) 4.3.c_h_s_y $3$ (not in LMFDB) 4.3.f_n_be_ci $3$ (not in LMFDB) 4.3.i_bc_ci_eb $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ag_r_abe_bw $2$ (not in LMFDB) 4.3.ae_h_a_am $2$ (not in LMFDB) 4.3.ac_b_ag_y $2$ (not in LMFDB) 4.3.a_ab_a_m $2$ (not in LMFDB) 4.3.c_b_g_y $2$ (not in LMFDB) 4.3.e_h_a_am $2$ (not in LMFDB) 4.3.g_r_be_bw $2$ (not in LMFDB) 4.3.k_bx_fu_ma $2$ (not in LMFDB) 4.3.ah_bc_acx_fu $3$ (not in LMFDB) 4.3.ae_e_m_abn $3$ (not in LMFDB) 4.3.ae_q_abk_da $3$ (not in LMFDB) 4.3.ab_b_g_am $3$ (not in LMFDB) 4.3.ab_e_d_g $3$ (not in LMFDB) 4.3.c_ac_a_p $3$ (not in LMFDB) 4.3.c_h_s_y $3$ (not in LMFDB) 4.3.f_n_be_ci $3$ (not in LMFDB) 4.3.i_bc_ci_eb $3$ (not in LMFDB) 4.3.ag_n_ag_am $4$ (not in LMFDB) 4.3.ae_n_ay_bw $4$ (not in LMFDB) 4.3.a_af_a_y $4$ (not in LMFDB) 4.3.a_b_a_m $4$ (not in LMFDB) 4.3.a_f_a_y $4$ (not in LMFDB) 4.3.e_n_y_bw $4$ (not in LMFDB) 4.3.g_n_g_am $4$ (not in LMFDB) 4.3.ai_bc_aci_eb $6$ (not in LMFDB) 4.3.af_n_abe_ci $6$ (not in LMFDB) 4.3.ad_i_ap_be $6$ (not in LMFDB) 4.3.ac_ac_a_p $6$ (not in LMFDB) 4.3.ac_h_as_y $6$ (not in LMFDB) 4.3.a_i_a_be $6$ (not in LMFDB) 4.3.b_b_ag_am $6$ (not in LMFDB) 4.3.b_e_ad_g $6$ (not in LMFDB) 4.3.d_i_p_be $6$ (not in LMFDB) 4.3.e_e_am_abn $6$ (not in LMFDB) 4.3.e_q_bk_da $6$ (not in LMFDB) 4.3.h_bc_cx_fu $6$ (not in LMFDB) 4.3.ak_bv_afi_kw $8$ (not in LMFDB) 4.3.ae_f_a_ag $8$ (not in LMFDB) 4.3.ae_l_ay_bq $8$ (not in LMFDB) 4.3.ac_ab_g_ag $8$ (not in LMFDB) 4.3.c_ab_ag_ag $8$ (not in LMFDB) 4.3.e_f_a_ag $8$ (not in LMFDB) 4.3.e_l_y_bq $8$ (not in LMFDB) 4.3.k_bv_fi_kw $8$ (not in LMFDB) 4.3.ae_e_m_abq $12$ (not in LMFDB) 4.3.ad_e_ad_g $12$ (not in LMFDB) 4.3.ac_af_g_m $12$ (not in LMFDB) 4.3.ac_e_am_v $12$ (not in LMFDB) 4.3.a_ai_a_be $12$ (not in LMFDB) 4.3.a_ae_a_g $12$ (not in LMFDB) 4.3.a_e_a_g $12$ (not in LMFDB) 4.3.c_af_ag_m $12$ (not in LMFDB) 4.3.c_e_m_v $12$ (not in LMFDB) 4.3.d_e_d_g $12$ (not in LMFDB) 4.3.e_e_am_abq $12$ (not in LMFDB) 4.3.ah_ba_acr_fi $24$ (not in LMFDB) 4.3.ae_c_m_abe $24$ (not in LMFDB) 4.3.ae_i_am_s $24$ (not in LMFDB) 4.3.ae_k_am_s $24$ (not in LMFDB) 4.3.ae_o_abk_co $24$ (not in LMFDB) 4.3.ac_b_ag_s $24$ (not in LMFDB) 4.3.ab_c_ad_ag $24$ (not in LMFDB) 4.3.a_ac_a_s $24$ (not in LMFDB) 4.3.a_c_a_s $24$ (not in LMFDB) 4.3.b_c_d_ag $24$ (not in LMFDB) 4.3.c_b_g_s $24$ (not in LMFDB) 4.3.e_c_am_abe $24$ (not in LMFDB) 4.3.e_i_m_s $24$ (not in LMFDB) 4.3.e_k_m_s $24$ (not in LMFDB) 4.3.e_o_bk_co $24$ (not in LMFDB) 4.3.h_ba_cr_fi $24$ (not in LMFDB)