# Properties

 Label 4.3.ak_bv_afi_kw 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0540867239847$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.445913276015$ 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 $A(\F_{q^r})$ 2 3332 490784 38291344 3619319362 327058457600 25217176717378 1880566937174016 147899391592223264 12058613378200922372

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 4 24 72 254 838 2402 6656 19392 58564

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 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.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.3.ae_i : $$\Q(\zeta_{8})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec 2 $\times$ 1.531441.zi 2 . The endomorphism algebra for each factor is: 1.531441.acec 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.531441.zi 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
All geometric endomorphisms are defined over $\F_{3^{12}}$.
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 $\times$ 2.9.a_ao. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.a_ao : $$\Q(\zeta_{8})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.ae_i. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.27.ae_i : $$\Q(\zeta_{8})$$.
• Endomorphism algebra over $\F_{3^{4}}$  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 2 $\times$ 1.81.j 2 . The endomorphism algebra for each factor is: 1.81.ao 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.81.j 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{3^{6}}$  The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 2.729.a_zi. The endomorphism algebra for each factor is: 1.729.cc 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.a_zi : $$\Q(\zeta_{8})$$.

## 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_f_a_ag $2$ (not in LMFDB) 4.3.ac_ab_g_ag $2$ (not in LMFDB) 4.3.c_ab_ag_ag $2$ (not in LMFDB) 4.3.e_f_a_ag $2$ (not in LMFDB) 4.3.k_bv_fi_kw $2$ (not in LMFDB) 4.3.ah_ba_acr_fi $3$ (not in LMFDB) 4.3.ae_o_abk_co $3$ (not in LMFDB) 4.3.ab_c_ad_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ae_f_a_ag $2$ (not in LMFDB) 4.3.ac_ab_g_ag $2$ (not in LMFDB) 4.3.c_ab_ag_ag $2$ (not in LMFDB) 4.3.e_f_a_ag $2$ (not in LMFDB) 4.3.k_bv_fi_kw $2$ (not in LMFDB) 4.3.ah_ba_acr_fi $3$ (not in LMFDB) 4.3.ae_o_abk_co $3$ (not in LMFDB) 4.3.ab_c_ad_ag $3$ (not in LMFDB) 4.3.ae_l_ay_bq $4$ (not in LMFDB) 4.3.e_l_y_bq $4$ (not in LMFDB) 4.3.b_c_d_ag $6$ (not in LMFDB) 4.3.e_o_bk_co $6$ (not in LMFDB) 4.3.h_ba_cr_fi $6$ (not in LMFDB) 4.3.ak_bx_afu_ma $8$ (not in LMFDB) 4.3.ag_n_ag_am $8$ (not in LMFDB) 4.3.ag_r_abe_bw $8$ (not in LMFDB) 4.3.ae_h_a_am $8$ (not in LMFDB) 4.3.ae_n_ay_bw $8$ (not in LMFDB) 4.3.ac_b_ag_y $8$ (not in LMFDB) 4.3.a_af_a_y $8$ (not in LMFDB) 4.3.a_ab_a_m $8$ (not in LMFDB) 4.3.a_b_a_m $8$ (not in LMFDB) 4.3.a_f_a_y $8$ (not in LMFDB) 4.3.c_b_g_y $8$ (not in LMFDB) 4.3.e_h_a_am $8$ (not in LMFDB) 4.3.e_n_y_bw $8$ (not in LMFDB) 4.3.g_n_g_am $8$ (not in LMFDB) 4.3.g_r_be_bw $8$ (not in LMFDB) 4.3.k_bx_fu_ma $8$ (not in LMFDB) 4.3.ae_c_m_abe $12$ (not in LMFDB) 4.3.e_c_am_abe $12$ (not in LMFDB) 4.3.ai_bc_aci_eb $24$ (not in LMFDB) 4.3.ah_bc_acx_fu $24$ (not in LMFDB) 4.3.af_n_abe_ci $24$ (not in LMFDB) 4.3.ae_e_m_abq $24$ (not in LMFDB) 4.3.ae_e_m_abn $24$ (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_q_abk_da $24$ (not in LMFDB) 4.3.ad_e_ad_g $24$ (not in LMFDB) 4.3.ad_i_ap_be $24$ (not in LMFDB) 4.3.ac_af_g_m $24$ (not in LMFDB) 4.3.ac_ac_a_p $24$ (not in LMFDB) 4.3.ac_b_ag_s $24$ (not in LMFDB) 4.3.ac_e_am_v $24$ (not in LMFDB) 4.3.ac_h_as_y $24$ (not in LMFDB) 4.3.ab_b_g_am $24$ (not in LMFDB) 4.3.ab_e_d_g $24$ (not in LMFDB) 4.3.a_ai_a_be $24$ (not in LMFDB) 4.3.a_ae_a_g $24$ (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.a_e_a_g $24$ (not in LMFDB) 4.3.a_i_a_be $24$ (not in LMFDB) 4.3.b_b_ag_am $24$ (not in LMFDB) 4.3.b_e_ad_g $24$ (not in LMFDB) 4.3.c_af_ag_m $24$ (not in LMFDB) 4.3.c_ac_a_p $24$ (not in LMFDB) 4.3.c_b_g_s $24$ (not in LMFDB) 4.3.c_e_m_v $24$ (not in LMFDB) 4.3.c_h_s_y $24$ (not in LMFDB) 4.3.d_e_d_g $24$ (not in LMFDB) 4.3.d_i_p_be $24$ (not in LMFDB) 4.3.e_e_am_abq $24$ (not in LMFDB) 4.3.e_e_am_abn $24$ (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_q_bk_da $24$ (not in LMFDB) 4.3.f_n_be_ci $24$ (not in LMFDB) 4.3.h_bc_cx_fu $24$ (not in LMFDB) 4.3.i_bc_ci_eb $24$ (not in LMFDB)