Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 5 x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.254551732336$, $\pm0.538152604671$
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
$A(\F_{q^r})$ 7 7889 680512 54662881 5079693647 343587786752 23073342565007 1827315445641777 150688359407419456 12108904870630602209

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 10 32 102 336 874 2208 6470 19760 58810

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ac_f 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.bk_bww. 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_i_am_v$2$(not in LMFDB)
4.3.ac_c_a_d$2$(not in LMFDB)
4.3.c_c_a_d$2$(not in LMFDB)
4.3.e_i_m_v$2$(not in LMFDB)
4.3.i_bg_dg_gj$2$(not in LMFDB)
4.3.af_r_abq_dg$3$(not in LMFDB)
4.3.ac_c_a_d$3$(not in LMFDB)
4.3.ac_l_as_bw$3$(not in LMFDB)
4.3.b_f_g_m$3$(not in LMFDB)
4.3.e_i_m_v$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.ae_i_am_v$2$(not in LMFDB)
4.3.ac_c_a_d$2$(not in LMFDB)
4.3.c_c_a_d$2$(not in LMFDB)
4.3.e_i_m_v$2$(not in LMFDB)
4.3.i_bg_dg_gj$2$(not in LMFDB)
4.3.af_r_abq_dg$3$(not in LMFDB)
4.3.ac_c_a_d$3$(not in LMFDB)
4.3.ac_l_as_bw$3$(not in LMFDB)
4.3.b_f_g_m$3$(not in LMFDB)
4.3.e_i_m_v$3$(not in LMFDB)
4.3.ac_i_am_bh$4$(not in LMFDB)
4.3.c_i_m_bh$4$(not in LMFDB)
4.3.ac_c_a_d$6$(not in LMFDB)
4.3.ab_f_ag_m$6$(not in LMFDB)
4.3.c_l_s_bw$6$(not in LMFDB)
4.3.f_r_bq_dg$6$(not in LMFDB)
4.3.ac_ab_g_am$12$(not in LMFDB)
4.3.c_ab_ag_am$12$(not in LMFDB)
4.3.ac_f_ag_s$24$(not in LMFDB)
4.3.c_f_g_s$24$(not in LMFDB)