# Properties

 Label 4.3.ai_bf_ada_fu 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 4 x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.210767374595$, $\pm0.567777800232$ Angle rank: $2$ (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 $A(\F_{q^r})$ 6 6468 550368 56840784 5473263966 351231648768 23674748355222 1880431371506688 150082676736082272 11912820414139488228

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 8 26 104 356 890 2264 6656 19682 57848

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ac_e 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_e : 4.0.7488.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 2.729.ca_czy. 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.ca_czy : 4.0.7488.1.
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 $\times$ 2.9.e_k. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.e_k : 4.0.7488.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.ac_bc. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.27.ac_bc : 4.0.7488.1.

## 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_h_ag_g $2$ (not in LMFDB) 4.3.ac_b_a_g $2$ (not in LMFDB) 4.3.c_b_a_g $2$ (not in LMFDB) 4.3.e_h_g_g $2$ (not in LMFDB) 4.3.i_bf_da_fu $2$ (not in LMFDB) 4.3.af_q_abn_da $3$ (not in LMFDB) 4.3.ac_b_a_g $3$ (not in LMFDB) 4.3.ac_k_as_bq $3$ (not in LMFDB) 4.3.b_e_d_g $3$ (not in LMFDB) 4.3.e_h_g_g $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ae_h_ag_g $2$ (not in LMFDB) 4.3.ac_b_a_g $2$ (not in LMFDB) 4.3.c_b_a_g $2$ (not in LMFDB) 4.3.e_h_g_g $2$ (not in LMFDB) 4.3.i_bf_da_fu $2$ (not in LMFDB) 4.3.af_q_abn_da $3$ (not in LMFDB) 4.3.ac_b_a_g $3$ (not in LMFDB) 4.3.ac_k_as_bq $3$ (not in LMFDB) 4.3.b_e_d_g $3$ (not in LMFDB) 4.3.e_h_g_g $3$ (not in LMFDB) 4.3.ac_h_am_be $4$ (not in LMFDB) 4.3.c_h_m_be $4$ (not in LMFDB) 4.3.af_q_abn_da $6$ (not in LMFDB) 4.3.ae_h_ag_g $6$ (not in LMFDB) 4.3.ac_b_a_g $6$ (not in LMFDB) 4.3.ac_k_as_bq $6$ (not in LMFDB) 4.3.ab_e_ad_g $6$ (not in LMFDB) 4.3.b_e_d_g $6$ (not in LMFDB) 4.3.c_b_a_g $6$ (not in LMFDB) 4.3.c_k_s_bq $6$ (not in LMFDB) 4.3.f_q_bn_da $6$ (not in LMFDB) 4.3.ac_ac_g_ag $12$ (not in LMFDB) 4.3.ac_h_am_be $12$ (not in LMFDB) 4.3.c_ac_ag_ag $12$ (not in LMFDB) 4.3.c_h_m_be $12$ (not in LMFDB) 4.3.ac_e_ag_s $24$ (not in LMFDB) 4.3.c_e_g_s $24$ (not in LMFDB)