# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $4$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0540867239847$, $\pm0.166666666667$, $\pm0.406785250661$, $\pm0.445913276015$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 6 7140 631008 31558800 2844704886 300359808000 25239404286102 1889625817497600 146877530321315808 12011040839153825700

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 12 32 56 196 774 2404 6688 19256 58332

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.ab $\times$ 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:
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.zi 2 $\times$ 1.531441.cag. The endomorphism algebra for each factor is: 1.531441.acec : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.531441.zi 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.531441.cag : $$\Q(\sqrt{-11})$$.
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 $\times$ 1.9.f $\times$ 2.9.a_ao. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.i $\times$ 2.27.ae_i. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{4}}$  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 2 $\times$ 1.81.ah $\times$ 1.81.j. The endomorphism algebra for each factor is: 1.81.ao 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.81.ah : $$\Q(\sqrt{-11})$$. 1.81.j : $$\Q(\sqrt{-3})$$.
• Endomorphism algebra over $\F_{3^{6}}$  The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak $\times$ 1.729.cc $\times$ 2.729.a_zi. The endomorphism algebra for each factor is: 1.729.ak : $$\Q(\sqrt{-11})$$. 1.729.cc : 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.ag_t_abu_dm $2$ (not in LMFDB) 4.3.ac_d_ac_ag $2$ (not in LMFDB) 4.3.a_b_ae_ag $2$ (not in LMFDB) 4.3.a_b_e_ag $2$ (not in LMFDB) 4.3.c_d_c_ag $2$ (not in LMFDB) 4.3.g_t_bu_dm $2$ (not in LMFDB) 4.3.i_bh_do_he $2$ (not in LMFDB) 4.3.af_s_abv_dm $3$ (not in LMFDB) 4.3.ac_d_ac_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ag_t_abu_dm $2$ (not in LMFDB) 4.3.ac_d_ac_ag $2$ (not in LMFDB) 4.3.a_b_ae_ag $2$ (not in LMFDB) 4.3.a_b_e_ag $2$ (not in LMFDB) 4.3.c_d_c_ag $2$ (not in LMFDB) 4.3.g_t_bu_dm $2$ (not in LMFDB) 4.3.i_bh_do_he $2$ (not in LMFDB) 4.3.af_s_abv_dm $3$ (not in LMFDB) 4.3.ac_d_ac_ag $3$ (not in LMFDB) 4.3.ad_k_az_bq $6$ (not in LMFDB) 4.3.d_k_z_bq $6$ (not in LMFDB) 4.3.f_s_bv_dm $6$ (not in LMFDB) 4.3.ai_bj_adw_hw $8$ (not in LMFDB) 4.3.ag_v_aby_ds $8$ (not in LMFDB) 4.3.ae_h_ae_a $8$ (not in LMFDB) 4.3.ae_l_au_bk $8$ (not in LMFDB) 4.3.ac_b_ac_m $8$ (not in LMFDB) 4.3.ac_f_ak_y $8$ (not in LMFDB) 4.3.ac_f_c_a $8$ (not in LMFDB) 4.3.a_d_ae_m $8$ (not in LMFDB) 4.3.a_d_e_m $8$ (not in LMFDB) 4.3.c_b_c_m $8$ (not in LMFDB) 4.3.c_f_ac_a $8$ (not in LMFDB) 4.3.c_f_k_y $8$ (not in LMFDB) 4.3.e_h_e_a $8$ (not in LMFDB) 4.3.e_l_u_bk $8$ (not in LMFDB) 4.3.g_v_by_ds $8$ (not in LMFDB) 4.3.i_bj_dw_hw $8$ (not in LMFDB) 4.3.ag_s_abo_cx $24$ (not in LMFDB) 4.3.af_u_abx_dy $24$ (not in LMFDB) 4.3.ae_i_au_bt $24$ (not in LMFDB) 4.3.ad_j_aw_bk $24$ (not in LMFDB) 4.3.ad_m_ax_cc $24$ (not in LMFDB) 4.3.ac_c_i_av $24$ (not in LMFDB) 4.3.ab_e_ab_g $24$ (not in LMFDB) 4.3.ab_f_ao_m $24$ (not in LMFDB) 4.3.ab_i_af_be $24$ (not in LMFDB) 4.3.a_a_ae_ad $24$ (not in LMFDB) 4.3.a_a_e_ad $24$ (not in LMFDB) 4.3.b_e_b_g $24$ (not in LMFDB) 4.3.b_f_o_m $24$ (not in LMFDB) 4.3.b_i_f_be $24$ (not in LMFDB) 4.3.c_c_ai_av $24$ (not in LMFDB) 4.3.d_j_w_bk $24$ (not in LMFDB) 4.3.d_m_x_cc $24$ (not in LMFDB) 4.3.e_i_u_bt $24$ (not in LMFDB) 4.3.f_u_bx_dy $24$ (not in LMFDB) 4.3.g_s_bo_cx $24$ (not in LMFDB)