Properties

Label 4.3.ah_w_abq_cr
Base field $\F_{3}$
Dimension $4$
$p$-rank $2$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - x + x^{2} - 3 x^{3} + 9 x^{4} )$
  $1 - 7 x + 22 x^{2} - 42 x^{3} + 69 x^{4} - 126 x^{5} + 198 x^{6} - 189 x^{7} + 81 x^{8}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.201748855633$, $\pm0.672988571819$
Angle rank:  $2$ (numerical)

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $7$ $5145$ $433552$ $73907925$ $5190222352$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-3$ $5$ $21$ $125$ $342$ $827$ $2433$ $6725$ $19173$ $58400$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ab_b 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.al_abfv. The endomorphism algebra for each factor is:
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.af_k_ag_ad$2$(not in LMFDB)
4.3.ab_ac_a_p$2$(not in LMFDB)
4.3.b_ac_a_p$2$(not in LMFDB)
4.3.f_k_g_ad$2$(not in LMFDB)
4.3.h_w_bq_cr$2$(not in LMFDB)
4.3.ae_k_av_bq$3$(not in LMFDB)
4.3.ab_ac_a_p$3$(not in LMFDB)
4.3.ab_h_aj_y$3$(not in LMFDB)
4.3.c_e_d_g$3$(not in LMFDB)
4.3.f_k_g_ad$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
4.3.af_k_ag_ad$2$(not in LMFDB)
4.3.ab_ac_a_p$2$(not in LMFDB)
4.3.b_ac_a_p$2$(not in LMFDB)
4.3.f_k_g_ad$2$(not in LMFDB)
4.3.h_w_bq_cr$2$(not in LMFDB)
4.3.ae_k_av_bq$3$(not in LMFDB)
4.3.ab_ac_a_p$3$(not in LMFDB)
4.3.ab_h_aj_y$3$(not in LMFDB)
4.3.c_e_d_g$3$(not in LMFDB)
4.3.f_k_g_ad$3$(not in LMFDB)
4.3.ab_e_ag_v$4$(not in LMFDB)
4.3.b_e_g_v$4$(not in LMFDB)
4.3.af_k_ag_ad$6$(not in LMFDB)
4.3.ae_k_av_bq$6$(not in LMFDB)
4.3.ac_e_ad_g$6$(not in LMFDB)
4.3.ab_ac_a_p$6$(not in LMFDB)
4.3.ab_h_aj_y$6$(not in LMFDB)
4.3.b_ac_a_p$6$(not in LMFDB)
4.3.b_h_j_y$6$(not in LMFDB)
4.3.c_e_d_g$6$(not in LMFDB)
4.3.e_k_v_bq$6$(not in LMFDB)
4.3.ab_af_d_m$12$(not in LMFDB)
4.3.ab_e_ag_v$12$(not in LMFDB)
4.3.b_af_ad_m$12$(not in LMFDB)
4.3.b_e_g_v$12$(not in LMFDB)
4.3.ab_b_ad_s$24$(not in LMFDB)
4.3.b_b_d_s$24$(not in LMFDB)