# Properties

 Label 5.3.ak_bt_aer_ix_apd Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 6 x^{2} - 7 x^{3} + 18 x^{4} - 36 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.0452398905210$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.239335307006$, $\pm0.691360448188$ Angle rank: $3$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 5 24255 8988560 5267337075 1082283492775 215575726544640 53420925605073745 12017202035056893675 2874085776613431618560 714887926089295297652775

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 0 15 116 304 765 2332 6484 19140 58800

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_g_ah 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}($$$\Q(\sqrt{-3})$$$)$ 3.3.ae_g_ah : 6.0.2461019.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 3.729.acv_eli_afacr. 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 $\infty$. 3.729.acv_eli_afacr : 6.0.2461019.1.
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 $\times$ 3.9.ae_q_acp. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.9.ae_q_acp : 6.0.2461019.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 3.27.an_bw_acv. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.27.an_bw_acv : 6.0.2461019.1.

## 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 5.3.ae_d_f_j_abz $2$ (not in LMFDB) 5.3.ac_ad_n_d_abn $2$ (not in LMFDB) 5.3.c_ad_an_d_bn $2$ (not in LMFDB) 5.3.e_d_af_j_bz $2$ (not in LMFDB) 5.3.k_bt_er_ix_pd $2$ (not in LMFDB) 5.3.ah_y_acg_eq_aio $3$ (not in LMFDB) 5.3.ae_d_f_j_abz $3$ (not in LMFDB) 5.3.ae_m_abf_cl_aek $3$ (not in LMFDB) 5.3.ab_a_ae_g_ag $3$ (not in LMFDB) 5.3.c_ad_an_d_bn $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ae_d_f_j_abz $2$ (not in LMFDB) 5.3.ac_ad_n_d_abn $2$ (not in LMFDB) 5.3.c_ad_an_d_bn $2$ (not in LMFDB) 5.3.e_d_af_j_bz $2$ (not in LMFDB) 5.3.k_bt_er_ix_pd $2$ (not in LMFDB) 5.3.ah_y_acg_eq_aio $3$ (not in LMFDB) 5.3.ae_d_f_j_abz $3$ (not in LMFDB) 5.3.ae_m_abf_cl_aek $3$ (not in LMFDB) 5.3.ab_a_ae_g_ag $3$ (not in LMFDB) 5.3.c_ad_an_d_bn $3$ (not in LMFDB) 5.3.ae_j_at_bt_adp $4$ (not in LMFDB) 5.3.e_j_t_bt_dp $4$ (not in LMFDB) 5.3.ah_y_acg_eq_aio $6$ (not in LMFDB) 5.3.ae_d_f_j_abz $6$ (not in LMFDB) 5.3.ae_m_abf_cl_aek $6$ (not in LMFDB) 5.3.ac_ad_n_d_abn $6$ (not in LMFDB) 5.3.ab_a_ae_g_ag $6$ (not in LMFDB) 5.3.b_a_e_g_g $6$ (not in LMFDB) 5.3.e_d_af_j_bz $6$ (not in LMFDB) 5.3.e_m_bf_cl_ek $6$ (not in LMFDB) 5.3.h_y_cg_eq_io $6$ (not in LMFDB) 5.3.ae_a_r_aj_abe $12$ (not in LMFDB) 5.3.ae_j_at_bt_adp $12$ (not in LMFDB) 5.3.e_a_ar_aj_be $12$ (not in LMFDB) 5.3.e_j_t_bt_dp $12$ (not in LMFDB) 5.3.ae_g_ah_bb_acu $24$ (not in LMFDB) 5.3.e_g_h_bb_cu $24$ (not in LMFDB)