# Properties

 Label 4.3.aj_bq_aev_jy 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: $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{3}$ Frobenius angles: $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$ Angle rank: $1$ (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 $A(\F_{q^r})$ 8 12096 1536416 80510976 3840744248 250890587136 21073303936952 1848011586011136 154514490015638816 12407390297278675776

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 13 58 133 265 646 2011 6541 20254 60253

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.ac 3 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 : $$\Q(\sqrt{-3})$$. 1.3.ac 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-2})$$$)$
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu 3 $\times$ 1.729.cc. The endomorphism algebra for each factor is: 1.729.abu 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-2})$$$)$ 1.729.cc : 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 $\times$ 1.9.c 3 . The endomorphism algebra for each factor is: 1.9.ad : $$\Q(\sqrt{-3})$$. 1.9.c 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-2})$$$)$
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.k 3 . The endomorphism algebra for each factor is: 1.27.a : $$\Q(\sqrt{-3})$$. 1.27.k 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-2})$$$)$

## 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.af_o_az_bq $2$ (not in LMFDB) 4.3.ad_g_b_ag $2$ (not in LMFDB) 4.3.ab_c_af_s $2$ (not in LMFDB) 4.3.b_c_f_s $2$ (not in LMFDB) 4.3.d_g_ab_ag $2$ (not in LMFDB) 4.3.f_o_z_bq $2$ (not in LMFDB) 4.3.j_bq_ev_jy $2$ (not in LMFDB) 4.3.ag_y_ack_ew $3$ (not in LMFDB) 4.3.ad_d_k_abe $3$ (not in LMFDB) 4.3.a_d_k_a $3$ (not in LMFDB) 4.3.d_d_k_be $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.af_o_az_bq $2$ (not in LMFDB) 4.3.ad_g_b_ag $2$ (not in LMFDB) 4.3.ab_c_af_s $2$ (not in LMFDB) 4.3.b_c_f_s $2$ (not in LMFDB) 4.3.d_g_ab_ag $2$ (not in LMFDB) 4.3.f_o_z_bq $2$ (not in LMFDB) 4.3.j_bq_ev_jy $2$ (not in LMFDB) 4.3.ag_y_ack_ew $3$ (not in LMFDB) 4.3.ad_d_k_abe $3$ (not in LMFDB) 4.3.a_d_k_a $3$ (not in LMFDB) 4.3.d_d_k_be $3$ (not in LMFDB) 4.3.af_k_af_ag $4$ (not in LMFDB) 4.3.ab_ac_ab_s $4$ (not in LMFDB) 4.3.b_ac_b_s $4$ (not in LMFDB) 4.3.f_k_f_ag $4$ (not in LMFDB) 4.3.ah_x_aby_dm $6$ (not in LMFDB) 4.3.ae_l_aba_bw $6$ (not in LMFDB) 4.3.ad_d_ak_be $6$ (not in LMFDB) 4.3.ac_i_ak_be $6$ (not in LMFDB) 4.3.ab_ab_ac_g $6$ (not in LMFDB) 4.3.a_d_ak_a $6$ (not in LMFDB) 4.3.b_ab_c_g $6$ (not in LMFDB) 4.3.c_i_k_be $6$ (not in LMFDB) 4.3.d_d_ak_abe $6$ (not in LMFDB) 4.3.e_l_ba_bw $6$ (not in LMFDB) 4.3.g_y_ck_ew $6$ (not in LMFDB) 4.3.h_x_by_dm $6$ (not in LMFDB) 4.3.aj_bo_ael_ja $8$ (not in LMFDB) 4.3.af_m_ax_bq $8$ (not in LMFDB) 4.3.ad_e_ab_ag $8$ (not in LMFDB) 4.3.ab_a_f_ag $8$ (not in LMFDB) 4.3.b_a_af_ag $8$ (not in LMFDB) 4.3.d_e_b_ag $8$ (not in LMFDB) 4.3.f_m_x_bq $8$ (not in LMFDB) 4.3.j_bo_el_ja $8$ (not in LMFDB) 4.3.ac_e_ac_g $12$ (not in LMFDB) 4.3.c_e_c_g $12$ (not in LMFDB) 4.3.ag_w_acg_ek $24$ (not in LMFDB) 4.3.ac_g_ao_s $24$ (not in LMFDB) 4.3.c_g_o_s $24$ (not in LMFDB) 4.3.g_w_cg_ek $24$ (not in LMFDB)