# Properties

 Label 5.3.al_cf_ahe_qz_abgf Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 + x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{4}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.593214749339$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 36015 12293120 5143122075 1483234632275 272017438801920 55794332176205705 12998950195866036675 2975480275028737387520 700532673766419908758575

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 3 20 111 383 936 2429 6999 19820 57603

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 4 $\times$ 1.3.b 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 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-3})$$$)$ 1.3.b : $$\Q(\sqrt{-11})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak $\times$ 1.729.cc 4 . The endomorphism algebra for each factor is: 1.729.ak : $$\Q(\sqrt{-11})$$. 1.729.cc 4 : $\mathrm{M}_{4}(B)$, 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 4 $\times$ 1.9.f. The endomorphism algebra for each factor is: 1.9.ad 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-3})$$$)$ 1.9.f : $$\Q(\sqrt{-11})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ai $\times$ 1.27.a 4 . The endomorphism algebra for each factor is: 1.27.ai : $$\Q(\sqrt{-11})$$. 1.27.a 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-3})$$$)$

## 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.an_dd_amg_bhp_acpn $2$ (not in LMFDB) 5.3.ah_v_abe_j_bb $2$ (not in LMFDB) 5.3.af_j_ag_j_abb $2$ (not in LMFDB) 5.3.ab_ad_g_j_abb $2$ (not in LMFDB) 5.3.b_ad_ag_j_bb $2$ (not in LMFDB) 5.3.f_j_g_j_bb $2$ (not in LMFDB) 5.3.h_v_be_j_abb $2$ (not in LMFDB) 5.3.l_cf_he_qz_bgf $2$ (not in LMFDB) 5.3.n_dd_mg_bhp_cpn $2$ (not in LMFDB) 5.3.ai_bh_ads_ir_aqq $3$ (not in LMFDB) 5.3.af_j_ag_j_abb $3$ (not in LMFDB) 5.3.af_s_abz_en_aii $3$ (not in LMFDB) 5.3.ac_d_ag_j_a $3$ (not in LMFDB) 5.3.ac_m_ay_cl_aee $3$ (not in LMFDB) 5.3.b_ad_ag_j_bb $3$ (not in LMFDB) 5.3.b_g_d_j_a $3$ (not in LMFDB) 5.3.b_p_m_dm_cc $3$ (not in LMFDB) 5.3.e_j_m_j_a $3$ (not in LMFDB) 5.3.e_s_bw_en_ii $3$ (not in LMFDB) 5.3.h_v_be_j_abb $3$ (not in LMFDB) 5.3.h_be_dp_ir_qq $3$ (not in LMFDB) 5.3.k_bz_gs_qz_bhg $3$ (not in LMFDB) 5.3.n_dd_mg_bhp_cpn $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.an_dd_amg_bhp_acpn $2$ (not in LMFDB) 5.3.ah_v_abe_j_bb $2$ (not in LMFDB) 5.3.af_j_ag_j_abb $2$ (not in LMFDB) 5.3.ab_ad_g_j_abb $2$ (not in LMFDB) 5.3.b_ad_ag_j_bb $2$ (not in LMFDB) 5.3.f_j_g_j_bb $2$ (not in LMFDB) 5.3.h_v_be_j_abb $2$ (not in LMFDB) 5.3.l_cf_he_qz_bgf $2$ (not in LMFDB) 5.3.n_dd_mg_bhp_cpn $2$ (not in LMFDB) 5.3.ai_bh_ads_ir_aqq $3$ (not in LMFDB) 5.3.af_j_ag_j_abb $3$ (not in LMFDB) 5.3.af_s_abz_en_aii $3$ (not in LMFDB) 5.3.ac_d_ag_j_a $3$ (not in LMFDB) 5.3.ac_m_ay_cl_aee $3$ (not in LMFDB) 5.3.b_ad_ag_j_bb $3$ (not in LMFDB) 5.3.b_g_d_j_a $3$ (not in LMFDB) 5.3.b_p_m_dm_cc $3$ (not in LMFDB) 5.3.e_j_m_j_a $3$ (not in LMFDB) 5.3.e_s_bw_en_ii $3$ (not in LMFDB) 5.3.h_v_be_j_abb $3$ (not in LMFDB) 5.3.h_be_dp_ir_qq $3$ (not in LMFDB) 5.3.k_bz_gs_qz_bhg $3$ (not in LMFDB) 5.3.n_dd_mg_bhp_cpn $3$ (not in LMFDB) 5.3.ah_bb_acu_fx_akt $4$ (not in LMFDB) 5.3.af_p_abk_dd_afx $4$ (not in LMFDB) 5.3.ab_d_a_j_aj $4$ (not in LMFDB) 5.3.ab_j_ag_bt_abb $4$ (not in LMFDB) 5.3.b_d_a_j_j $4$ (not in LMFDB) 5.3.b_j_g_bt_bb $4$ (not in LMFDB) 5.3.f_p_bk_dd_fx $4$ (not in LMFDB) 5.3.h_bb_cu_fx_kt $4$ (not in LMFDB) 5.3.e_m_y_bk_cl $5$ (not in LMFDB) 5.3.ak_bz_ags_qz_abhg $6$ (not in LMFDB) 5.3.ah_be_adp_ir_aqq $6$ (not in LMFDB) 5.3.ae_j_am_j_a $6$ (not in LMFDB) 5.3.ae_s_abw_en_aii $6$ (not in LMFDB) 5.3.ab_g_ad_j_a $6$ (not in LMFDB) 5.3.ab_p_am_dm_acc $6$ (not in LMFDB) 5.3.c_d_g_j_a $6$ (not in LMFDB) 5.3.c_m_y_cl_ee $6$ (not in LMFDB) 5.3.f_s_bz_en_ii $6$ (not in LMFDB) 5.3.i_bh_ds_ir_qq $6$ (not in LMFDB) 5.3.ab_d_a_aj_j $8$ (not in LMFDB) 5.3.b_d_a_aj_aj $8$ (not in LMFDB) 5.3.ac_d_ap_bb_abb $9$ (not in LMFDB) 5.3.ac_d_d_aj_bb $9$ (not in LMFDB) 5.3.b_g_ag_a_acc $9$ (not in LMFDB) 5.3.b_g_m_s_cc $9$ (not in LMFDB) 5.3.e_j_d_abb_add $9$ (not in LMFDB) 5.3.e_j_v_bt_dd $9$ (not in LMFDB) 5.3.ae_m_ay_bk_acl $10$ (not in LMFDB) 5.3.ac_g_am_s_abt $10$ (not in LMFDB) 5.3.c_g_m_s_bt $10$ (not in LMFDB) 5.3.ah_s_aj_acl_gy $12$ (not in LMFDB) 5.3.af_g_j_abb_bk $12$ (not in LMFDB) 5.3.ae_g_a_abb_cu $12$ (not in LMFDB) 5.3.ae_p_abk_dd_afo $12$ (not in LMFDB) 5.3.ac_a_a_aj_bk $12$ (not in LMFDB) 5.3.ac_j_as_bt_acu $12$ (not in LMFDB) 5.3.ab_aj_m_s_acc $12$ (not in LMFDB) 5.3.ab_ag_j_j_abk $12$ (not in LMFDB) 5.3.ab_a_d_aj_a $12$ (not in LMFDB) 5.3.ab_d_a_as_s $12$ (not in LMFDB) 5.3.ab_m_aj_cl_abk $12$ (not in LMFDB) 5.3.b_aj_am_s_cc $12$ (not in LMFDB) 5.3.b_ag_aj_j_bk $12$ (not in LMFDB) 5.3.b_a_ad_aj_a $12$ (not in LMFDB) 5.3.b_d_a_as_as $12$ (not in LMFDB) 5.3.b_m_j_cl_bk $12$ (not in LMFDB) 5.3.c_a_a_aj_abk $12$ (not in LMFDB) 5.3.c_j_s_bt_cu $12$ (not in LMFDB) 5.3.e_g_a_abb_acu $12$ (not in LMFDB) 5.3.e_p_bk_dd_fo $12$ (not in LMFDB) 5.3.f_g_aj_abb_abk $12$ (not in LMFDB) 5.3.h_s_j_acl_agy $12$ (not in LMFDB) 5.3.ac_g_am_s_abt $15$ (not in LMFDB) 5.3.b_a_ad_a_j $15$ (not in LMFDB) 5.3.ae_j_av_bt_add $18$ (not in LMFDB) 5.3.ae_j_ad_abb_dd $18$ (not in LMFDB) 5.3.ab_g_am_s_acc $18$ (not in LMFDB) 5.3.ab_g_g_a_cc $18$ (not in LMFDB) 5.3.c_d_ad_aj_abb $18$ (not in LMFDB) 5.3.c_d_p_bb_bb $18$ (not in LMFDB) 5.3.ah_y_abz_dd_aew $24$ (not in LMFDB) 5.3.af_m_av_bt_adm $24$ (not in LMFDB) 5.3.ae_m_ay_bt_acu $24$ (not in LMFDB) 5.3.ac_g_am_bb_abk $24$ (not in LMFDB) 5.3.ab_ad_g_a_as $24$ (not in LMFDB) 5.3.ab_a_d_j_as $24$ (not in LMFDB) 5.3.ab_d_a_s_as $24$ (not in LMFDB) 5.3.ab_g_ad_bb_as $24$ (not in LMFDB) 5.3.ab_j_ag_bk_as $24$ (not in LMFDB) 5.3.b_ad_ag_a_s $24$ (not in LMFDB) 5.3.b_a_ad_j_s $24$ (not in LMFDB) 5.3.b_d_a_s_s $24$ (not in LMFDB) 5.3.b_g_d_bb_s $24$ (not in LMFDB) 5.3.b_j_g_bk_s $24$ (not in LMFDB) 5.3.c_g_m_bb_bk $24$ (not in LMFDB) 5.3.e_m_y_bt_cu $24$ (not in LMFDB) 5.3.f_m_v_bt_dm $24$ (not in LMFDB) 5.3.h_y_bz_dd_ew $24$ (not in LMFDB) 5.3.ab_a_d_a_aj $30$ (not in LMFDB) 5.3.ab_d_a_a_a $48$ (not in LMFDB) 5.3.b_d_a_a_a $48$ (not in LMFDB) 5.3.ab_g_ad_s_aj $60$ (not in LMFDB) 5.3.b_g_d_s_j $60$ (not in LMFDB)