# Properties

 Label 3.13.ap_dy_aqx Base Field $\F_{13}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{13}$ Dimension: $3$ L-polynomial: $( 1 - x + 13 x^{2} )( 1 - 7 x + 13 x^{2} )^{2}$ Frobenius angles: $\pm0.0772104791556$, $\pm0.0772104791556$, $\pm0.455715642762$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

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

## 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 $A(\F_{q^r})$ 637 4213755 10125466624 22758933199275 50961221941086457 112500615380537180160 247131956265713730738709 542811441823314176232085275 1192537920497238107784957242368 2620019459632019140624088299414275

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 149 2096 27893 369659 4828748 62765639 815746757 10604540528 137859744989

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah 2 $\times$ 1.13.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.13.ah 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.13.ab : $$\Q(\sqrt{-51})$$.
All geometric endomorphisms are defined over $\F_{13}$.

## 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 3.13.an_cw_ald $2$ (not in LMFDB) 3.13.ab_ak_x $2$ (not in LMFDB) 3.13.b_ak_ax $2$ (not in LMFDB) 3.13.n_cw_ld $2$ (not in LMFDB) 3.13.p_dy_qx $2$ (not in LMFDB) 3.13.ag_be_afm $3$ (not in LMFDB) 3.13.ad_g_abr $3$ (not in LMFDB) 3.13.d_bn_cw $3$ (not in LMFDB) 3.13.g_bq_fq $3$ (not in LMFDB) 3.13.j_cc_ib $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.an_cw_ald $2$ (not in LMFDB) 3.13.ab_ak_x $2$ (not in LMFDB) 3.13.b_ak_ax $2$ (not in LMFDB) 3.13.n_cw_ld $2$ (not in LMFDB) 3.13.p_dy_qx $2$ (not in LMFDB) 3.13.ag_be_afm $3$ (not in LMFDB) 3.13.ad_g_abr $3$ (not in LMFDB) 3.13.d_bn_cw $3$ (not in LMFDB) 3.13.g_bq_fq $3$ (not in LMFDB) 3.13.j_cc_ib $3$ (not in LMFDB) 3.13.ab_bk_ax $4$ (not in LMFDB) 3.13.b_bk_x $4$ (not in LMFDB) 3.13.an_di_aoj $6$ (not in LMFDB) 3.13.al_ck_ajr $6$ (not in LMFDB) 3.13.al_cw_alz $6$ (not in LMFDB) 3.13.ak_ck_ako $6$ (not in LMFDB) 3.13.aj_cc_aib $6$ (not in LMFDB) 3.13.ai_bs_ahm $6$ (not in LMFDB) 3.13.ai_ce_aik $6$ (not in LMFDB) 3.13.ag_bq_afq $6$ (not in LMFDB) 3.13.af_bv_afe $6$ (not in LMFDB) 3.13.ae_u_aeo $6$ (not in LMFDB) 3.13.ae_bg_adq $6$ (not in LMFDB) 3.13.ad_bn_acw $6$ (not in LMFDB) 3.13.ac_ba_ack $6$ (not in LMFDB) 3.13.ab_c_acj $6$ (not in LMFDB) 3.13.ab_o_ab $6$ (not in LMFDB) 3.13.ab_bj_aw $6$ (not in LMFDB) 3.13.b_c_cj $6$ (not in LMFDB) 3.13.b_o_b $6$ (not in LMFDB) 3.13.b_bj_w $6$ (not in LMFDB) 3.13.c_ba_ck $6$ (not in LMFDB) 3.13.d_g_br $6$ (not in LMFDB) 3.13.e_u_eo $6$ (not in LMFDB) 3.13.e_bg_dq $6$ (not in LMFDB) 3.13.f_bv_fe $6$ (not in LMFDB) 3.13.g_be_fm $6$ (not in LMFDB) 3.13.i_bs_hm $6$ (not in LMFDB) 3.13.i_ce_ik $6$ (not in LMFDB) 3.13.k_ck_ko $6$ (not in LMFDB) 3.13.l_ck_jr $6$ (not in LMFDB) 3.13.l_cw_lz $6$ (not in LMFDB) 3.13.n_di_oj $6$ (not in LMFDB) 3.13.ab_aj_w $12$ (not in LMFDB) 3.13.ab_m_b $12$ (not in LMFDB) 3.13.b_aj_aw $12$ (not in LMFDB) 3.13.b_m_ab $12$ (not in LMFDB)