# Properties

 Label 4.3.aj_bo_aek_ix 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 3 x + 7 x^{2} - 9 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.227267020856$, $\pm0.464830336654$ 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 $A(\F_{q^r})$ 5 7105 827120 54033525 4553342000 357302606080 25077434754755 1831615948705725 146237929245768080 12037965374815072000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 9 37 101 310 903 2389 6485 19171 58464

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ad_h 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.ad_h : 4.0.1525.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 2.729.cn_dov. 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.cn_dov : 4.0.1525.1.
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.f_n. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.f_n : 4.0.1525.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.j_cv. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.27.j_cv : 4.0.1525.1.

## 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.ad_e_ag_p $2$ (not in LMFDB) 4.3.ad_e_a_ad $2$ (not in LMFDB) 4.3.d_e_a_ad $2$ (not in LMFDB) 4.3.d_e_g_p $2$ (not in LMFDB) 4.3.j_bo_ek_ix $2$ (not in LMFDB) 4.3.ag_w_acf_ek $3$ (not in LMFDB) 4.3.ad_n_abb_ci $3$ (not in LMFDB) 4.3.a_e_d_g $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ad_e_ag_p $2$ (not in LMFDB) 4.3.ad_e_a_ad $2$ (not in LMFDB) 4.3.d_e_a_ad $2$ (not in LMFDB) 4.3.d_e_g_p $2$ (not in LMFDB) 4.3.j_bo_ek_ix $2$ (not in LMFDB) 4.3.ag_w_acf_ek $3$ (not in LMFDB) 4.3.ad_n_abb_ci $3$ (not in LMFDB) 4.3.a_e_d_g $3$ (not in LMFDB) 4.3.ad_k_as_bn $4$ (not in LMFDB) 4.3.d_k_s_bn $4$ (not in LMFDB) 4.3.a_e_ad_g $6$ (not in LMFDB) 4.3.d_n_bb_ci $6$ (not in LMFDB) 4.3.g_w_cf_ek $6$ (not in LMFDB) 4.3.ad_b_j_ay $12$ (not in LMFDB) 4.3.d_b_aj_ay $12$ (not in LMFDB) 4.3.ad_h_aj_s $24$ (not in LMFDB) 4.3.d_h_j_s $24$ (not in LMFDB)