# Properties

 Label 4.3.ak_bw_afo_ll Base Field $\F_{3}$ Dimension $4$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

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

This isogeny class is not simple.

## Newton polygon

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

## 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 $A(\F_{q^r})$ 3 5145 790272 56517825 4239234843 346961018880 26528984248539 1956789130794225 149087900677992192 11912402839226961225

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 6 36 102 294 882 2514 6918 19548 57846

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 1.3.ab 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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.3.ab : $$\Q(\sqrt{-11})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak $\times$ 1.729.cc 3 . The endomorphism algebra for each factor is: 1.729.ak : $$\Q(\sqrt{-11})$$. 1.729.cc 3 : $\mathrm{M}_{3}(B)$, where $B$ is 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 3 $\times$ 1.9.f. The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.9.f : $$\Q(\sqrt{-11})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 1.27.i. The endomorphism algebra for each factor is: 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.27.i : $$\Q(\sqrt{-11})$$.

## 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.ai_be_acu_ff $2$ (not in LMFDB) 4.3.ac_a_a_j $2$ (not in LMFDB) 4.3.e_g_a_aj $2$ (not in LMFDB) 4.3.k_bw_fo_ll $2$ (not in LMFDB) 4.3.ah_bb_acu_fo $3$ (not in LMFDB) 4.3.ae_g_a_aj $3$ (not in LMFDB) 4.3.ae_p_abk_cu $3$ (not in LMFDB) 4.3.ab_d_a_a $3$ (not in LMFDB) 4.3.ab_m_aj_cc $3$ (not in LMFDB) 4.3.c_a_a_j $3$ (not in LMFDB) 4.3.c_j_s_bk $3$ (not in LMFDB) 4.3.f_p_bk_cu $3$ (not in LMFDB) 4.3.i_be_cu_ff $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ai_be_acu_ff $2$ (not in LMFDB) 4.3.ac_a_a_j $2$ (not in LMFDB) 4.3.e_g_a_aj $2$ (not in LMFDB) 4.3.k_bw_fo_ll $2$ (not in LMFDB) 4.3.ah_bb_acu_fo $3$ (not in LMFDB) 4.3.ae_g_a_aj $3$ (not in LMFDB) 4.3.ae_p_abk_cu $3$ (not in LMFDB) 4.3.ab_d_a_a $3$ (not in LMFDB) 4.3.ab_m_aj_cc $3$ (not in LMFDB) 4.3.c_a_a_j $3$ (not in LMFDB) 4.3.c_j_s_bk $3$ (not in LMFDB) 4.3.f_p_bk_cu $3$ (not in LMFDB) 4.3.i_be_cu_ff $3$ (not in LMFDB) 4.3.ae_m_ay_bt $4$ (not in LMFDB) 4.3.ac_g_am_bb $4$ (not in LMFDB) 4.3.c_g_m_bb $4$ (not in LMFDB) 4.3.e_m_y_bt $4$ (not in LMFDB) 4.3.af_p_abk_cu $6$ (not in LMFDB) 4.3.ac_j_as_bk $6$ (not in LMFDB) 4.3.b_d_a_a $6$ (not in LMFDB) 4.3.b_m_j_cc $6$ (not in LMFDB) 4.3.e_p_bk_cu $6$ (not in LMFDB) 4.3.h_bb_cu_fo $6$ (not in LMFDB) 4.3.ab_d_aj_j $9$ (not in LMFDB) 4.3.ab_d_j_aj $9$ (not in LMFDB) 4.3.ae_d_m_abk $12$ (not in LMFDB) 4.3.ac_ad_g_a $12$ (not in LMFDB) 4.3.ab_a_d_as $12$ (not in LMFDB) 4.3.ab_j_ag_bk $12$ (not in LMFDB) 4.3.b_a_ad_as $12$ (not in LMFDB) 4.3.b_j_g_bk $12$ (not in LMFDB) 4.3.c_ad_ag_a $12$ (not in LMFDB) 4.3.e_d_am_abk $12$ (not in LMFDB) 4.3.ab_d_j_aj $18$ (not in LMFDB) 4.3.b_d_aj_aj $18$ (not in LMFDB) 4.3.b_d_j_j $18$ (not in LMFDB) 4.3.ae_j_am_s $24$ (not in LMFDB) 4.3.ac_d_ag_s $24$ (not in LMFDB) 4.3.ab_g_ad_s $24$ (not in LMFDB) 4.3.b_g_d_s $24$ (not in LMFDB) 4.3.c_d_g_s $24$ (not in LMFDB) 4.3.e_j_m_s $24$ (not in LMFDB)