# Properties

 Label 5.3.am_cr_ajs_zh_abxw Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{4}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.5$ Angle rank: $0$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

## 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 $A(\F_{q^r})$ 4 38416 17210368 4388797504 1316033637364 296196766695424 57994298718070348 12463412921879046400 2955063255987167847424 712025834662527967355536

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -8 4 28 100 352 1000 2512 6724 19684 58564

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 4 $\times$ 1.3.a 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.a : $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 5 and its endomorphism algebra is $\mathrm{M}_{5}(B)$, 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.g. The endomorphism algebra for each factor is: 1.9.ad 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-3})$$$)$ 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 5 and its endomorphism algebra is $\mathrm{M}_{5}($$$\Q(\sqrt{-3})$$$)$

## 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.ag_p_as_j_a $2$ (not in LMFDB) 5.3.a_ad_a_j_a $2$ (not in LMFDB) 5.3.g_p_s_j_a $2$ (not in LMFDB) 5.3.m_cr_js_zh_bxw $2$ (not in LMFDB) 5.3.ap_eb_ari_byf_adyv $3$ (not in LMFDB) 5.3.aj_bh_acc_j_dd $3$ (not in LMFDB) 5.3.aj_bq_aff_mv_ayy $3$ (not in LMFDB) 5.3.ag_p_as_j_a $3$ (not in LMFDB) 5.3.ag_y_acu_gp_amm $3$ (not in LMFDB) 5.3.ad_ad_s_j_add $3$ (not in LMFDB) 5.3.ad_g_aj_j_a $3$ (not in LMFDB) 5.3.ad_p_abk_dm_agg $3$ (not in LMFDB) 5.3.a_ad_a_j_a $3$ (not in LMFDB) 5.3.a_g_a_j_a $3$ (not in LMFDB) 5.3.a_p_a_dm_a $3$ (not in LMFDB) 5.3.d_ad_as_j_dd $3$ (not in LMFDB) 5.3.d_g_j_j_a $3$ (not in LMFDB) 5.3.d_p_bk_dm_gg $3$ (not in LMFDB) 5.3.g_p_s_j_a $3$ (not in LMFDB) 5.3.g_y_cu_gp_mm $3$ (not in LMFDB) 5.3.j_bh_cc_j_add $3$ (not in LMFDB) 5.3.j_bq_ff_mv_yy $3$ (not in LMFDB) 5.3.m_cr_js_zh_bxw $3$ (not in LMFDB) 5.3.p_eb_ri_byf_dyv $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ag_p_as_j_a $2$ (not in LMFDB) 5.3.a_ad_a_j_a $2$ (not in LMFDB) 5.3.g_p_s_j_a $2$ (not in LMFDB) 5.3.m_cr_js_zh_bxw $2$ (not in LMFDB) 5.3.ap_eb_ari_byf_adyv $3$ (not in LMFDB) 5.3.aj_bh_acc_j_dd $3$ (not in LMFDB) 5.3.aj_bq_aff_mv_ayy $3$ (not in LMFDB) 5.3.ag_p_as_j_a $3$ (not in LMFDB) 5.3.ag_y_acu_gp_amm $3$ (not in LMFDB) 5.3.ad_ad_s_j_add $3$ (not in LMFDB) 5.3.ad_g_aj_j_a $3$ (not in LMFDB) 5.3.ad_p_abk_dm_agg $3$ (not in LMFDB) 5.3.a_ad_a_j_a $3$ (not in LMFDB) 5.3.a_g_a_j_a $3$ (not in LMFDB) 5.3.a_p_a_dm_a $3$ (not in LMFDB) 5.3.d_ad_as_j_dd $3$ (not in LMFDB) 5.3.d_g_j_j_a $3$ (not in LMFDB) 5.3.d_p_bk_dm_gg $3$ (not in LMFDB) 5.3.g_p_s_j_a $3$ (not in LMFDB) 5.3.g_y_cu_gp_mm $3$ (not in LMFDB) 5.3.j_bh_cc_j_add $3$ (not in LMFDB) 5.3.j_bq_ff_mv_yy $3$ (not in LMFDB) 5.3.m_cr_js_zh_bxw $3$ (not in LMFDB) 5.3.p_eb_ri_byf_dyv $3$ (not in LMFDB) 5.3.ag_v_acc_en_aii $4$ (not in LMFDB) 5.3.a_d_a_j_a $4$ (not in LMFDB) 5.3.a_j_a_bt_a $4$ (not in LMFDB) 5.3.g_v_cc_en_ii $4$ (not in LMFDB) 5.3.d_j_s_bb_cc $5$ (not in LMFDB) 5.3.a_d_a_aj_a $8$ (not in LMFDB) 5.3.ag_p_abb_cl_aff $9$ (not in LMFDB) 5.3.ag_p_aj_abt_ff $9$ (not in LMFDB) 5.3.ad_g_as_bk_acc $9$ (not in LMFDB) 5.3.ad_g_a_as_cc $9$ (not in LMFDB) 5.3.a_ad_aj_j_bb $9$ (not in LMFDB) 5.3.a_ad_j_j_abb $9$ (not in LMFDB) 5.3.a_g_aj_j_acc $9$ (not in LMFDB) 5.3.a_g_j_j_cc $9$ (not in LMFDB) 5.3.d_g_a_as_acc $9$ (not in LMFDB) 5.3.d_g_s_bk_cc $9$ (not in LMFDB) 5.3.g_p_j_abt_aff $9$ (not in LMFDB) 5.3.g_p_bb_cl_ff $9$ (not in LMFDB) 5.3.ad_j_as_bb_acc $10$ (not in LMFDB) 5.3.aj_be_abb_adv_mm $12$ (not in LMFDB) 5.3.aj_bn_aee_ir_app $12$ (not in LMFDB) 5.3.ag_m_a_abt_ee $12$ (not in LMFDB) 5.3.ad_aj_bk_s_agg $12$ (not in LMFDB) 5.3.ad_ag_bb_j_aee $12$ (not in LMFDB) 5.3.ad_a_j_aj_a $12$ (not in LMFDB) 5.3.ad_d_a_as_cc $12$ (not in LMFDB) 5.3.ad_d_a_j_abb $12$ (not in LMFDB) 5.3.ad_j_as_bt_add $12$ (not in LMFDB) 5.3.ad_m_abb_cl_aee $12$ (not in LMFDB) 5.3.a_aj_a_s_a $12$ (not in LMFDB) 5.3.a_ag_a_j_a $12$ (not in LMFDB) 5.3.a_a_a_aj_a $12$ (not in LMFDB) 5.3.a_d_a_as_a $12$ (not in LMFDB) 5.3.a_m_a_cl_a $12$ (not in LMFDB) 5.3.d_aj_abk_s_gg $12$ (not in LMFDB) 5.3.d_ag_abb_j_ee $12$ (not in LMFDB) 5.3.d_a_aj_aj_a $12$ (not in LMFDB) 5.3.d_d_a_as_acc $12$ (not in LMFDB) 5.3.d_d_a_j_bb $12$ (not in LMFDB) 5.3.d_j_s_bt_dd $12$ (not in LMFDB) 5.3.d_m_bb_cl_ee $12$ (not in LMFDB) 5.3.g_m_a_abt_aee $12$ (not in LMFDB) 5.3.j_be_bb_adv_amm $12$ (not in LMFDB) 5.3.j_bn_ee_ir_pp $12$ (not in LMFDB) 5.3.ag_s_abk_cc_add $15$ (not in LMFDB) 5.3.ad_a_j_a_abb $15$ (not in LMFDB) 5.3.ad_j_as_bb_acc $15$ (not in LMFDB) 5.3.a_a_a_a_abb $15$ (not in LMFDB) 5.3.a_a_a_a_a $15$ (not in LMFDB) 5.3.a_a_a_a_bb $15$ (not in LMFDB) 5.3.d_a_aj_a_bb $15$ (not in LMFDB) 5.3.g_s_bk_cc_dd $15$ (not in LMFDB) 5.3.aj_bk_add_en_agg $24$ (not in LMFDB) 5.3.ag_s_abk_cl_aee $24$ (not in LMFDB) 5.3.ad_ad_s_a_acc $24$ (not in LMFDB) 5.3.ad_a_j_j_acc $24$ (not in LMFDB) 5.3.ad_d_a_aj_bb $24$ (not in LMFDB) 5.3.ad_d_a_s_acc $24$ (not in LMFDB) 5.3.ad_g_aj_bb_acc $24$ (not in LMFDB) 5.3.ad_j_as_bk_acc $24$ (not in LMFDB) 5.3.a_ad_a_a_a $24$ (not in LMFDB) 5.3.a_a_a_j_a $24$ (not in LMFDB) 5.3.a_d_a_s_a $24$ (not in LMFDB) 5.3.a_g_a_bb_a $24$ (not in LMFDB) 5.3.a_j_a_bk_a $24$ (not in LMFDB) 5.3.d_ad_as_a_cc $24$ (not in LMFDB) 5.3.d_a_aj_j_cc $24$ (not in LMFDB) 5.3.d_d_a_aj_abb $24$ (not in LMFDB) 5.3.d_d_a_s_cc $24$ (not in LMFDB) 5.3.d_g_j_bb_cc $24$ (not in LMFDB) 5.3.d_j_s_bk_cc $24$ (not in LMFDB) 5.3.g_s_bk_cl_ee $24$ (not in LMFDB) 5.3.j_bk_dd_en_gg $24$ (not in LMFDB) 5.3.a_ag_aj_j_cc $36$ (not in LMFDB) 5.3.a_ag_j_j_acc $36$ (not in LMFDB) 5.3.a_d_aj_j_abb $36$ (not in LMFDB) 5.3.a_d_j_j_bb $36$ (not in LMFDB) 5.3.ad_d_a_a_a $48$ (not in LMFDB) 5.3.a_d_a_a_a $48$ (not in LMFDB) 5.3.d_d_a_a_a $48$ (not in LMFDB) 5.3.ad_g_aj_s_abb $60$ (not in LMFDB) 5.3.a_g_a_s_a $60$ (not in LMFDB) 5.3.d_g_j_s_bb $60$ (not in LMFDB) 5.3.a_a_aj_j_a $72$ (not in LMFDB) 5.3.a_a_j_j_a $72$ (not in LMFDB)