# Properties

 Label 5.3.ak_bx_agb_ol_abbp 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 10 x^{2} - 19 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.125412673718$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.335294135736$, $\pm0.584823404300$ 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 $A(\F_{q^r})$ 9 56007 14881104 4514892291 1174205479779 232624036802304 52709453025633189 13118037723941971275 3065690147851241616384 713523065769308099282607

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 8 27 100 324 821 2304 7060 20412 58688

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_k_at 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_k_at : 6.0.11822771.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 3.729.ar_fe_abgaf. 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.ar_fe_abgaf : 6.0.11822771.1.
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.e_i_f. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.9.e_i_f : 6.0.11822771.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 3.27.ab_ai_jr. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.27.ab_ai_jr : 6.0.11822771.1.

## 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_h_ah_j_ap $2$ (not in LMFDB) 5.3.ac_b_b_d_ad $2$ (not in LMFDB) 5.3.c_b_ab_d_d $2$ (not in LMFDB) 5.3.e_h_h_j_p $2$ (not in LMFDB) 5.3.k_bx_gb_ol_bbp $2$ (not in LMFDB) 5.3.ah_bc_ade_hk_aoc $3$ (not in LMFDB) 5.3.ae_h_ah_j_ap $3$ (not in LMFDB) 5.3.ae_q_abr_dv_ahe $3$ (not in LMFDB) 5.3.ab_e_ae_g_ag $3$ (not in LMFDB) 5.3.c_b_ab_d_d $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ae_h_ah_j_ap $2$ (not in LMFDB) 5.3.ac_b_b_d_ad $2$ (not in LMFDB) 5.3.c_b_ab_d_d $2$ (not in LMFDB) 5.3.e_h_h_j_p $2$ (not in LMFDB) 5.3.k_bx_gb_ol_bbp $2$ (not in LMFDB) 5.3.ah_bc_ade_hk_aoc $3$ (not in LMFDB) 5.3.ae_h_ah_j_ap $3$ (not in LMFDB) 5.3.ae_q_abr_dv_ahe $3$ (not in LMFDB) 5.3.ab_e_ae_g_ag $3$ (not in LMFDB) 5.3.c_b_ab_d_d $3$ (not in LMFDB) 5.3.ae_n_abf_cr_aez $4$ (not in LMFDB) 5.3.e_n_bf_cr_ez $4$ (not in LMFDB) 5.3.ah_bc_ade_hk_aoc $6$ (not in LMFDB) 5.3.ae_h_ah_j_ap $6$ (not in LMFDB) 5.3.ae_q_abr_dv_ahe $6$ (not in LMFDB) 5.3.ac_b_b_d_ad $6$ (not in LMFDB) 5.3.ab_e_ae_g_ag $6$ (not in LMFDB) 5.3.b_e_e_g_g $6$ (not in LMFDB) 5.3.e_h_h_j_p $6$ (not in LMFDB) 5.3.e_q_br_dv_he $6$ (not in LMFDB) 5.3.h_bc_de_hk_oc $6$ (not in LMFDB) 5.3.ae_e_f_av_bq $12$ (not in LMFDB) 5.3.ae_n_abf_cr_aez $12$ (not in LMFDB) 5.3.e_e_af_av_abq $12$ (not in LMFDB) 5.3.e_n_bf_cr_ez $12$ (not in LMFDB) 5.3.ae_k_at_bn_acu $24$ (not in LMFDB) 5.3.e_k_t_bn_cu $24$ (not in LMFDB)