# Properties

 Label 3.13.aq_et_avo 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: $( 1 - 4 x + 13 x^{2} )( 1 - 6 x + 13 x^{2} )^{2}$ Frobenius angles: $\pm0.187167041811$, $\pm0.187167041811$, $\pm0.312832958189$ Angle rank: $1$ (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 $A(\F_{q^r})$ 640 4608000 11245402240 23887872000000 51482189428931200 112550220068921856000 247082989636358136952960 542801726578599395328000000 1192527299149168267745577554560 2619982662302134663644776256000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 160 2326 29276 373438 4830880 62753206 815732156 10604446078 137857808800

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ag 2 $\times$ 1.13.ae 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.ag 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.13.ae : $$\Q(\sqrt{-1})$$.
Endomorphism algebra over $\overline{\F}_{13}$
 The base change of $A$ to $\F_{13^{4}}$ is 1.28561.je 3 and its endomorphism algebra is $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{13^{2}}$  The base change of $A$ to $\F_{13^{2}}$ is 1.169.ak 2 $\times$ 1.169.k. The endomorphism algebra for each factor is: 1.169.ak 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.169.k : $$\Q(\sqrt{-1})$$.

## 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 3.13.ai_bb_acm $2$ (not in LMFDB) 3.13.ae_d_bo $2$ (not in LMFDB) 3.13.e_d_abo $2$ (not in LMFDB) 3.13.i_bb_cm $2$ (not in LMFDB) 3.13.q_et_vo $2$ (not in LMFDB) 3.13.c_m_cm $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ai_bb_acm $2$ (not in LMFDB) 3.13.ae_d_bo $2$ (not in LMFDB) 3.13.e_d_abo $2$ (not in LMFDB) 3.13.i_bb_cm $2$ (not in LMFDB) 3.13.q_et_vo $2$ (not in LMFDB) 3.13.c_m_cm $3$ (not in LMFDB) 3.13.as_fr_abai $4$ (not in LMFDB) 3.13.ao_dz_ars $4$ (not in LMFDB) 3.13.am_dj_aom $4$ (not in LMFDB) 3.13.ag_d_ci $4$ (not in LMFDB) 3.13.ag_x_aci $4$ (not in LMFDB) 3.13.ae_x_abo $4$ (not in LMFDB) 3.13.ac_h_bs $4$ (not in LMFDB) 3.13.c_h_abs $4$ (not in LMFDB) 3.13.e_x_bo $4$ (not in LMFDB) 3.13.g_d_aci $4$ (not in LMFDB) 3.13.g_x_ci $4$ (not in LMFDB) 3.13.m_dj_om $4$ (not in LMFDB) 3.13.o_dz_rs $4$ (not in LMFDB) 3.13.s_fr_bai $4$ (not in LMFDB) 3.13.ak_ci_ajo $6$ (not in LMFDB) 3.13.ac_m_acm $6$ (not in LMFDB) 3.13.k_ci_jo $6$ (not in LMFDB) 3.13.ag_al_fo $8$ (not in LMFDB) 3.13.ag_bl_afo $8$ (not in LMFDB) 3.13.ae_al_ds $8$ (not in LMFDB) 3.13.ae_bl_ads $8$ (not in LMFDB) 3.13.e_al_ads $8$ (not in LMFDB) 3.13.e_bl_ds $8$ (not in LMFDB) 3.13.g_al_afo $8$ (not in LMFDB) 3.13.g_bl_fo $8$ (not in LMFDB) 3.13.am_cu_ali $12$ (not in LMFDB) 3.13.ak_bo_aes $12$ (not in LMFDB) 3.13.ai_bg_aem $12$ (not in LMFDB) 3.13.ac_ai_di $12$ (not in LMFDB) 3.13.a_a_ado $12$ (not in LMFDB) 3.13.a_a_as $12$ (not in LMFDB) 3.13.a_a_s $12$ (not in LMFDB) 3.13.a_a_do $12$ (not in LMFDB) 3.13.c_ai_adi $12$ (not in LMFDB) 3.13.i_bg_em $12$ (not in LMFDB) 3.13.k_bo_es $12$ (not in LMFDB) 3.13.m_cu_li $12$ (not in LMFDB)