# Properties

 Label 4.3.aj_bp_aeq_jm 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: $( 1 - 2 x + 3 x^{2} )( 1 - x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.406785250661$ 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 8820 1072512 59623200 3785589786 302705786880 24576432300762 1922914262505600 150526973078358912 12059264472339338100

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 11 46 107 265 782 2347 6803 19738 58571

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 1.3.ac $\times$ 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 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.3.ac : $$\Q(\sqrt{-2})$$. 1.3.ab : $$\Q(\sqrt{-11})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.ak $\times$ 1.729.cc 2 . The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.ak : $$\Q(\sqrt{-11})$$. 1.729.cc 2 : $\mathrm{M}_{2}(B)$, 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 2 $\times$ 1.9.c $\times$ 1.9.f. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.9.c : $$\Q(\sqrt{-2})$$. 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 2 $\times$ 1.27.i $\times$ 1.27.k. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.27.i : $$\Q(\sqrt{-11})$$. 1.27.k : $$\Q(\sqrt{-2})$$.

## 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.ah_z_aci_ek $2$ (not in LMFDB) 4.3.af_n_ay_bq $2$ (not in LMFDB) 4.3.ad_f_am_be $2$ (not in LMFDB) 4.3.ab_b_a_g $2$ (not in LMFDB) 4.3.b_b_a_g $2$ (not in LMFDB) 4.3.d_f_a_ag $2$ (not in LMFDB) 4.3.f_n_y_bq $2$ (not in LMFDB) 4.3.h_z_ci_ek $2$ (not in LMFDB) 4.3.j_bp_eq_jm $2$ (not in LMFDB) 4.3.ag_x_aci_eq $3$ (not in LMFDB) 4.3.ad_f_a_ag $3$ (not in LMFDB) 4.3.ad_o_abb_co $3$ (not in LMFDB) 4.3.a_f_g_m $3$ (not in LMFDB) 4.3.d_f_m_be $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ah_z_aci_ek $2$ (not in LMFDB) 4.3.af_n_ay_bq $2$ (not in LMFDB) 4.3.ad_f_am_be $2$ (not in LMFDB) 4.3.ab_b_a_g $2$ (not in LMFDB) 4.3.b_b_a_g $2$ (not in LMFDB) 4.3.d_f_a_ag $2$ (not in LMFDB) 4.3.f_n_y_bq $2$ (not in LMFDB) 4.3.h_z_ci_ek $2$ (not in LMFDB) 4.3.j_bp_eq_jm $2$ (not in LMFDB) 4.3.ag_x_aci_eq $3$ (not in LMFDB) 4.3.ad_f_a_ag $3$ (not in LMFDB) 4.3.ad_o_abb_co $3$ (not in LMFDB) 4.3.a_f_g_m $3$ (not in LMFDB) 4.3.d_f_m_be $3$ (not in LMFDB) 4.3.ad_l_as_bq $4$ (not in LMFDB) 4.3.ab_h_ag_be $4$ (not in LMFDB) 4.3.b_h_g_be $4$ (not in LMFDB) 4.3.d_l_s_bq $4$ (not in LMFDB) 4.3.ae_n_abe_ci $6$ (not in LMFDB) 4.3.ad_o_abb_co $6$ (not in LMFDB) 4.3.ac_h_am_y $6$ (not in LMFDB) 4.3.ab_k_aj_bq $6$ (not in LMFDB) 4.3.a_f_ag_m $6$ (not in LMFDB) 4.3.b_k_j_bq $6$ (not in LMFDB) 4.3.c_h_m_y $6$ (not in LMFDB) 4.3.d_o_bb_co $6$ (not in LMFDB) 4.3.e_n_be_ci $6$ (not in LMFDB) 4.3.g_x_ci_eq $6$ (not in LMFDB) 4.3.ad_c_j_abe $12$ (not in LMFDB) 4.3.ab_ac_d_ag $12$ (not in LMFDB) 4.3.b_ac_ad_ag $12$ (not in LMFDB) 4.3.d_c_aj_abe $12$ (not in LMFDB) 4.3.ad_i_aj_s $24$ (not in LMFDB) 4.3.ab_e_ad_s $24$ (not in LMFDB) 4.3.b_e_d_s $24$ (not in LMFDB) 4.3.d_i_j_s $24$ (not in LMFDB)