Properties

Label 4.3.ak_bw_afo_ll
Base Field $\F_{3}$
Dimension $4$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{3}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.406785250661$
Angle rank:  $1$ (numerical)

This isogeny class is not simple.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1]$

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
$A(\F_{q^r})$ 3 5145 790272 56517825 4239234843 346961018880 26528984248539 1956789130794225 149087900677992192 11912402839226961225

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -6 6 36 102 294 882 2514 6918 19548 57846

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 3 $\times$ 1.3.ab 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 $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak $\times$ 1.729.cc 3 . The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{6}}$.
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
4.3.ai_be_acu_ff$2$(not in LMFDB)
4.3.ac_a_a_j$2$(not in LMFDB)
4.3.e_g_a_aj$2$(not in LMFDB)
4.3.k_bw_fo_ll$2$(not in LMFDB)
4.3.ah_bb_acu_fo$3$(not in LMFDB)
4.3.ae_g_a_aj$3$(not in LMFDB)
4.3.ae_p_abk_cu$3$(not in LMFDB)
4.3.ab_d_a_a$3$(not in LMFDB)
4.3.ab_m_aj_cc$3$(not in LMFDB)
4.3.c_a_a_j$3$(not in LMFDB)
4.3.c_j_s_bk$3$(not in LMFDB)
4.3.f_p_bk_cu$3$(not in LMFDB)
4.3.i_be_cu_ff$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ai_be_acu_ff$2$(not in LMFDB)
4.3.ac_a_a_j$2$(not in LMFDB)
4.3.e_g_a_aj$2$(not in LMFDB)
4.3.k_bw_fo_ll$2$(not in LMFDB)
4.3.ah_bb_acu_fo$3$(not in LMFDB)
4.3.ae_g_a_aj$3$(not in LMFDB)
4.3.ae_p_abk_cu$3$(not in LMFDB)
4.3.ab_d_a_a$3$(not in LMFDB)
4.3.ab_m_aj_cc$3$(not in LMFDB)
4.3.c_a_a_j$3$(not in LMFDB)
4.3.c_j_s_bk$3$(not in LMFDB)
4.3.f_p_bk_cu$3$(not in LMFDB)
4.3.i_be_cu_ff$3$(not in LMFDB)
4.3.ae_m_ay_bt$4$(not in LMFDB)
4.3.ac_g_am_bb$4$(not in LMFDB)
4.3.c_g_m_bb$4$(not in LMFDB)
4.3.e_m_y_bt$4$(not in LMFDB)
4.3.af_p_abk_cu$6$(not in LMFDB)
4.3.ac_j_as_bk$6$(not in LMFDB)
4.3.b_d_a_a$6$(not in LMFDB)
4.3.b_m_j_cc$6$(not in LMFDB)
4.3.e_p_bk_cu$6$(not in LMFDB)
4.3.h_bb_cu_fo$6$(not in LMFDB)
4.3.ab_d_aj_j$9$(not in LMFDB)
4.3.ab_d_j_aj$9$(not in LMFDB)
4.3.ae_d_m_abk$12$(not in LMFDB)
4.3.ac_ad_g_a$12$(not in LMFDB)
4.3.ab_a_d_as$12$(not in LMFDB)
4.3.ab_j_ag_bk$12$(not in LMFDB)
4.3.b_a_ad_as$12$(not in LMFDB)
4.3.b_j_g_bk$12$(not in LMFDB)
4.3.c_ad_ag_a$12$(not in LMFDB)
4.3.e_d_am_abk$12$(not in LMFDB)
4.3.ab_d_j_aj$18$(not in LMFDB)
4.3.b_d_aj_aj$18$(not in LMFDB)
4.3.b_d_j_j$18$(not in LMFDB)
4.3.ae_j_am_s$24$(not in LMFDB)
4.3.ac_d_ag_s$24$(not in LMFDB)
4.3.ab_g_ad_s$24$(not in LMFDB)
4.3.b_g_d_s$24$(not in LMFDB)
4.3.c_d_g_s$24$(not in LMFDB)
4.3.e_j_m_s$24$(not in LMFDB)