# Properties

 Label 4.3.aj_bm_ady_ht 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0975263560046$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.527857038681$ 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 $A(\F_{q^r})$ 3 3969 430416 40916421 4586243568 355330789632 24599121127689 1899813952225917 153645866448077808 12221700189956054784

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 5 19 77 310 899 2347 6725 20143 59360

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ad_f 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_f : 4.0.2197.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 2.729.cj_ddt. 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.cj_ddt : 4.0.2197.1.
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.b_al. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.b_al : 4.0.2197.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.aj_ct. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.27.aj_ct : 4.0.2197.1.

## 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_c_a_d $2$ (not in LMFDB) 4.3.ad_c_g_ap $2$ (not in LMFDB) 4.3.d_c_ag_ap $2$ (not in LMFDB) 4.3.d_c_a_d $2$ (not in LMFDB) 4.3.j_bm_dy_ht $2$ (not in LMFDB) 4.3.ag_u_abz_dy $3$ (not in LMFDB) 4.3.ad_c_a_d $3$ (not in LMFDB) 4.3.ad_l_abb_bw $3$ (not in LMFDB) 4.3.a_c_ad_ag $3$ (not in LMFDB) 4.3.d_c_ag_ap $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ad_c_a_d $2$ (not in LMFDB) 4.3.ad_c_g_ap $2$ (not in LMFDB) 4.3.d_c_ag_ap $2$ (not in LMFDB) 4.3.d_c_a_d $2$ (not in LMFDB) 4.3.j_bm_dy_ht $2$ (not in LMFDB) 4.3.ag_u_abz_dy $3$ (not in LMFDB) 4.3.ad_c_a_d $3$ (not in LMFDB) 4.3.ad_l_abb_bw $3$ (not in LMFDB) 4.3.a_c_ad_ag $3$ (not in LMFDB) 4.3.d_c_ag_ap $3$ (not in LMFDB) 4.3.ad_i_as_bh $4$ (not in LMFDB) 4.3.d_i_s_bh $4$ (not in LMFDB) 4.3.a_c_d_ag $6$ (not in LMFDB) 4.3.d_l_bb_bw $6$ (not in LMFDB) 4.3.g_u_bz_dy $6$ (not in LMFDB) 4.3.ad_ab_j_am $12$ (not in LMFDB) 4.3.d_ab_aj_am $12$ (not in LMFDB) 4.3.ad_f_aj_s $24$ (not in LMFDB) 4.3.d_f_j_s $24$ (not in LMFDB)