Properties

Label 3.9.aq_ei_aqw
Base field $\F_{3^{2}}$
Dimension $3$
$p$-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:  $\F_{3^{2}}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x )^{2}( 1 - 5 x + 9 x^{2} )^{2}$
  $1 - 16 x + 112 x^{2} - 438 x^{3} + 1008 x^{4} - 1296 x^{5} + 729 x^{6}$
Frobenius angles:  $0$, $0$, $\pm0.186429498677$, $\pm0.186429498677$
Angle rank:  $1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $100$ $360000$ $370177600$ $285156000000$ $207505465502500$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-6$ $50$ $696$ $6626$ $59514$ $532700$ $4783626$ $43034306$ $387339384$ $3486451250$

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 $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3^{2}}$
The isogeny class factors as 1.9.ag $\times$ 1.9.af 2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
3.9.ag_c_bq$2$(not in LMFDB)
3.9.ae_ai_da$2$(not in LMFDB)
3.9.e_ai_ada$2$(not in LMFDB)
3.9.g_c_abq$2$(not in LMFDB)
3.9.q_ei_qw$2$(not in LMFDB)
3.9.ah_w_abz$3$(not in LMFDB)
3.9.ab_af_ag$3$(not in LMFDB)
3.9.i_bo_fi$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.9.ag_c_bq$2$(not in LMFDB)
3.9.ae_ai_da$2$(not in LMFDB)
3.9.e_ai_ada$2$(not in LMFDB)
3.9.g_c_abq$2$(not in LMFDB)
3.9.q_ei_qw$2$(not in LMFDB)
3.9.ah_w_abz$3$(not in LMFDB)
3.9.ab_af_ag$3$(not in LMFDB)
3.9.i_bo_fi$3$(not in LMFDB)
3.9.ak_ca_agy$4$(not in LMFDB)
3.9.ag_q_abq$4$(not in LMFDB)
3.9.a_c_a$4$(not in LMFDB)
3.9.a_q_a$4$(not in LMFDB)
3.9.g_q_bq$4$(not in LMFDB)
3.9.k_ca_gy$4$(not in LMFDB)
3.9.an_de_alx$6$(not in LMFDB)
3.9.al_cd_ahe$6$(not in LMFDB)
3.9.ai_bo_afi$6$(not in LMFDB)
3.9.ad_c_v$6$(not in LMFDB)
3.9.ac_k_abq$6$(not in LMFDB)
3.9.b_af_g$6$(not in LMFDB)
3.9.c_k_bq$6$(not in LMFDB)
3.9.d_c_av$6$(not in LMFDB)
3.9.h_w_bz$6$(not in LMFDB)
3.9.l_cd_he$6$(not in LMFDB)
3.9.n_de_lx$6$(not in LMFDB)
3.9.af_z_adm$12$(not in LMFDB)
3.9.ad_q_av$12$(not in LMFDB)
3.9.d_q_v$12$(not in LMFDB)
3.9.f_z_dm$12$(not in LMFDB)