# Properties

 Label 4.3.ai_bi_adr_hk 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: $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 7 x^{2} - 9 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.227267020856$, $\pm0.304086723985$, $\pm0.464830336654$ Angle rank: $3$ (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 $A(\F_{q^r})$ 10 12180 1122520 57002400 4066084000 311728294080 23231717873290 1799908010409600 147649492269375880 12186375001164672000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 14 47 106 281 803 2222 6370 19361 59189

## Decomposition and endomorphism algebra

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

## 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_k_at_bk $2$ (not in LMFDB) 4.3.ac_e_af_m $2$ (not in LMFDB) 4.3.ac_e_b_a $2$ (not in LMFDB) 4.3.c_e_ab_a $2$ (not in LMFDB) 4.3.c_e_f_m $2$ (not in LMFDB) 4.3.e_k_t_bk $2$ (not in LMFDB) 4.3.i_bi_dr_hk $2$ (not in LMFDB) 4.3.af_t_abv_ds $3$ (not in LMFDB) 4.3.ac_e_b_a $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ae_k_at_bk $2$ (not in LMFDB) 4.3.ac_e_af_m $2$ (not in LMFDB) 4.3.ac_e_b_a $2$ (not in LMFDB) 4.3.c_e_ab_a $2$ (not in LMFDB) 4.3.c_e_f_m $2$ (not in LMFDB) 4.3.e_k_t_bk $2$ (not in LMFDB) 4.3.i_bi_dr_hk $2$ (not in LMFDB) 4.3.af_t_abv_ds $3$ (not in LMFDB) 4.3.ac_e_b_a $3$ (not in LMFDB) 4.3.ab_h_ah_y $6$ (not in LMFDB) 4.3.b_h_h_y $6$ (not in LMFDB) 4.3.f_t_bv_ds $6$ (not in LMFDB)