# Properties

 Label 5.3.al_ch_ahw_tt_abml Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 - 3 x + 3 x^{2} )^{3}( 1 - 2 x + 5 x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.254551732336$, $\pm0.538152604671$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 $A(\F_{q^r})$ 7 55223 19054336 4974322171 1376596978337 269372824813568 52353414280000883 12138856505398324611 2966149666575644571904 712088368727173824104663

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 7 32 111 363 928 2289 6551 19760 58567

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 2.3.ac_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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.3.ac_f : 4.0.4672.2.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 3 $\times$ 2.729.bk_bww. The endomorphism algebra for each factor is: 1.729.cc 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.bk_bww : 4.0.4672.2.
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 3 $\times$ 2.9.g_t. The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.9.g_t : 4.0.4672.2.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 2.27.e_ba. The endomorphism algebra for each factor is: 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.27.e_ba : 4.0.4672.2.

## 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 5.3.ah_x_abw_dd_aff $2$ (not in LMFDB) 5.3.af_l_am_j_aj $2$ (not in LMFDB) 5.3.ab_ab_a_j_aj $2$ (not in LMFDB) 5.3.b_ab_a_j_j $2$ (not in LMFDB) 5.3.f_l_m_j_j $2$ (not in LMFDB) 5.3.h_x_bw_dd_ff $2$ (not in LMFDB) 5.3.l_ch_hw_tt_bml $2$ (not in LMFDB) 5.3.ai_bj_aee_kb_atk $3$ (not in LMFDB) 5.3.af_l_am_j_aj $3$ (not in LMFDB) 5.3.af_u_acf_ff_ajs $3$ (not in LMFDB) 5.3.ac_f_ag_j_a $3$ (not in LMFDB) 5.3.ac_o_ay_dd_aee $3$ (not in LMFDB) 5.3.b_ab_a_j_j $3$ (not in LMFDB) 5.3.b_i_j_bb_bk $3$ (not in LMFDB) 5.3.e_l_y_bt_cu $3$ (not in LMFDB) 5.3.h_x_bw_dd_ff $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ah_x_abw_dd_aff $2$ (not in LMFDB) 5.3.af_l_am_j_aj $2$ (not in LMFDB) 5.3.ab_ab_a_j_aj $2$ (not in LMFDB) 5.3.b_ab_a_j_j $2$ (not in LMFDB) 5.3.f_l_m_j_j $2$ (not in LMFDB) 5.3.h_x_bw_dd_ff $2$ (not in LMFDB) 5.3.l_ch_hw_tt_bml $2$ (not in LMFDB) 5.3.ai_bj_aee_kb_atk $3$ (not in LMFDB) 5.3.af_l_am_j_aj $3$ (not in LMFDB) 5.3.af_u_acf_ff_ajs $3$ (not in LMFDB) 5.3.ac_f_ag_j_a $3$ (not in LMFDB) 5.3.ac_o_ay_dd_aee $3$ (not in LMFDB) 5.3.b_ab_a_j_j $3$ (not in LMFDB) 5.3.b_i_j_bb_bk $3$ (not in LMFDB) 5.3.e_l_y_bt_cu $3$ (not in LMFDB) 5.3.h_x_bw_dd_ff $3$ (not in LMFDB) 5.3.af_r_abq_dp_agp $4$ (not in LMFDB) 5.3.ab_f_ag_v_abb $4$ (not in LMFDB) 5.3.b_f_g_v_bb $4$ (not in LMFDB) 5.3.f_r_bq_dp_gp $4$ (not in LMFDB) 5.3.ae_l_ay_bt_acu $6$ (not in LMFDB) 5.3.ab_i_aj_bb_abk $6$ (not in LMFDB) 5.3.c_f_g_j_a $6$ (not in LMFDB) 5.3.c_o_y_dd_ee $6$ (not in LMFDB) 5.3.f_u_cf_ff_js $6$ (not in LMFDB) 5.3.i_bj_ee_kb_tk $6$ (not in LMFDB) 5.3.ac_f_ap_bb_abt $9$ (not in LMFDB) 5.3.ac_f_d_aj_bt $9$ (not in LMFDB) 5.3.af_i_d_abh_cu $12$ (not in LMFDB) 5.3.ac_c_a_ap_bk $12$ (not in LMFDB) 5.3.ac_l_as_cf_acu $12$ (not in LMFDB) 5.3.ab_ae_d_d_a $12$ (not in LMFDB) 5.3.b_ae_ad_d_a $12$ (not in LMFDB) 5.3.c_c_a_ap_abk $12$ (not in LMFDB) 5.3.c_l_s_cf_cu $12$ (not in LMFDB) 5.3.f_i_ad_abh_acu $12$ (not in LMFDB) 5.3.c_f_ad_aj_abt $18$ (not in LMFDB) 5.3.c_f_p_bb_bt $18$ (not in LMFDB) 5.3.af_o_abb_bz_adm $24$ (not in LMFDB) 5.3.ac_i_am_bh_abk $24$ (not in LMFDB) 5.3.ab_c_ad_p_as $24$ (not in LMFDB) 5.3.b_c_d_p_s $24$ (not in LMFDB) 5.3.c_i_m_bh_bk $24$ (not in LMFDB) 5.3.f_o_bb_bz_dm $24$ (not in LMFDB)