Properties

Label 4.3.ai_bf_ada_fu
Base Field $\F_{3}$
Dimension $4$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 4 x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.210767374595$, $\pm0.567777800232$
Angle rank:  $2$ (numerical)

This isogeny class is not simple.

Newton polygon

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

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
$A(\F_{q^r})$ 6 6468 550368 56840784 5473263966 351231648768 23674748355222 1880431371506688 150082676736082272 11912820414139488228

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 8 26 104 356 890 2264 6656 19682 57848

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ac_e 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.cc 2 $\times$ 2.729.ca_czy. 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.ae_h_ag_g$2$(not in LMFDB)
4.3.ac_b_a_g$2$(not in LMFDB)
4.3.c_b_a_g$2$(not in LMFDB)
4.3.e_h_g_g$2$(not in LMFDB)
4.3.i_bf_da_fu$2$(not in LMFDB)
4.3.af_q_abn_da$3$(not in LMFDB)
4.3.ac_b_a_g$3$(not in LMFDB)
4.3.ac_k_as_bq$3$(not in LMFDB)
4.3.b_e_d_g$3$(not in LMFDB)
4.3.e_h_g_g$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ae_h_ag_g$2$(not in LMFDB)
4.3.ac_b_a_g$2$(not in LMFDB)
4.3.c_b_a_g$2$(not in LMFDB)
4.3.e_h_g_g$2$(not in LMFDB)
4.3.i_bf_da_fu$2$(not in LMFDB)
4.3.af_q_abn_da$3$(not in LMFDB)
4.3.ac_b_a_g$3$(not in LMFDB)
4.3.ac_k_as_bq$3$(not in LMFDB)
4.3.b_e_d_g$3$(not in LMFDB)
4.3.e_h_g_g$3$(not in LMFDB)
4.3.ac_h_am_be$4$(not in LMFDB)
4.3.c_h_m_be$4$(not in LMFDB)
4.3.af_q_abn_da$6$(not in LMFDB)
4.3.ae_h_ag_g$6$(not in LMFDB)
4.3.ac_b_a_g$6$(not in LMFDB)
4.3.ac_k_as_bq$6$(not in LMFDB)
4.3.ab_e_ad_g$6$(not in LMFDB)
4.3.b_e_d_g$6$(not in LMFDB)
4.3.c_b_a_g$6$(not in LMFDB)
4.3.c_k_s_bq$6$(not in LMFDB)
4.3.f_q_bn_da$6$(not in LMFDB)
4.3.ac_ac_g_ag$12$(not in LMFDB)
4.3.ac_h_am_be$12$(not in LMFDB)
4.3.c_ac_ag_ag$12$(not in LMFDB)
4.3.c_h_m_be$12$(not in LMFDB)
4.3.ac_e_ag_s$24$(not in LMFDB)
4.3.c_e_g_s$24$(not in LMFDB)