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

Learn more about

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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 640 4608000 11245402240 23887872000000 51482189428931200 112550220068921856000 247082989636358136952960 542801726578599395328000000 1192527299149168267745577554560 2619982662302134663644776256000000

Point counts of the (virtual) curve

$r$ 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:
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

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)