# Properties

 Label 4.3.ai_bd_aco_eq 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.116139763599$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.616139763599$ 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 3920 341824 52998400 5070660404 326947819520 24919076318756 1992815309721600 151936029738601024 12058002552079778000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 4 14 96 336 838 2376 7040 19922 58564

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ac_c 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.ac_c : $$\Q(i, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec 2 $\times$ 1.531441.sk 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.sk 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$
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_ac. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.a_ac : $$\Q(i, \sqrt{5})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.ao_du. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.27.ao_du : $$\Q(i, \sqrt{5})$$.
• Endomorphism algebra over $\F_{3^{4}}$  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac 2 $\times$ 1.81.j 2 . The endomorphism algebra for each factor is: 1.81.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$ 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_sk. 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_sk : $$\Q(i, \sqrt{5})$$.

## 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_g_ay $2$ (not in LMFDB) 4.3.c_ab_a_m $2$ (not in LMFDB) 4.3.i_bd_co_eq $2$ (not in LMFDB) 4.3.af_o_abh_co $3$ (not in LMFDB) 4.3.ac_ab_a_m $3$ (not in LMFDB) 4.3.ac_i_as_be $3$ (not in LMFDB) 4.3.b_c_ad_ag $3$ (not in LMFDB) 4.3.e_f_ag_ay $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ae_f_g_ay $2$ (not in LMFDB) 4.3.c_ab_a_m $2$ (not in LMFDB) 4.3.i_bd_co_eq $2$ (not in LMFDB) 4.3.af_o_abh_co $3$ (not in LMFDB) 4.3.ac_ab_a_m $3$ (not in LMFDB) 4.3.ac_i_as_be $3$ (not in LMFDB) 4.3.b_c_ad_ag $3$ (not in LMFDB) 4.3.e_f_ag_ay $3$ (not in LMFDB) 4.3.ac_f_am_y $4$ (not in LMFDB) 4.3.c_f_m_y $4$ (not in LMFDB) 4.3.ab_c_d_ag $6$ (not in LMFDB) 4.3.b_c_ad_ag $6$ (not in LMFDB) 4.3.c_i_s_be $6$ (not in LMFDB) 4.3.f_o_bh_co $6$ (not in LMFDB) 4.3.ag_l_g_abq $8$ (not in LMFDB) 4.3.ag_t_abq_da $8$ (not in LMFDB) 4.3.a_ah_a_be $8$ (not in LMFDB) 4.3.a_ab_a_g $8$ (not in LMFDB) 4.3.a_b_a_g $8$ (not in LMFDB) 4.3.a_h_a_be $8$ (not in LMFDB) 4.3.g_l_ag_abq $8$ (not in LMFDB) 4.3.g_t_bq_da $8$ (not in LMFDB) 4.3.ac_ae_g_g $12$ (not in LMFDB) 4.3.c_ae_ag_g $12$ (not in LMFDB) 4.3.ad_c_d_ag $24$ (not in LMFDB) 4.3.ad_k_av_bq $24$ (not in LMFDB) 4.3.ac_c_ag_s $24$ (not in LMFDB) 4.3.a_ak_a_bq $24$ (not in LMFDB) 4.3.a_ae_a_s $24$ (not in LMFDB) 4.3.a_ac_a_ag $24$ (not in LMFDB) 4.3.a_c_a_ag $24$ (not in LMFDB) 4.3.a_e_a_s $24$ (not in LMFDB) 4.3.a_k_a_bq $24$ (not in LMFDB) 4.3.c_c_g_s $24$ (not in LMFDB) 4.3.d_c_ad_ag $24$ (not in LMFDB) 4.3.d_k_v_bq $24$ (not in LMFDB)