Properties

Label 3.9.ao_dn_ank
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:  $( 1 - 3 x )^{2}( 1 - 4 x + 9 x^{2} )^{2}$
  $1 - 14 x + 91 x^{2} - 348 x^{3} + 819 x^{4} - 1134 x^{5} + 729 x^{6}$
Frobenius angles:  $0$, $0$, $\pm0.267720472801$, $\pm0.267720472801$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $144$ $451584$ $404975376$ $289013760000$ $205842492172944$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-4$ $68$ $764$ $6716$ $59036$ $529028$ $4770524$ $43009916$ $387359036$ $3486791108$

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.ae 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_l_am$2$(not in LMFDB)
3.9.ac_af_ci$2$(not in LMFDB)
3.9.c_af_aci$2$(not in LMFDB)
3.9.g_l_m$2$(not in LMFDB)
3.9.o_dn_nk$2$(not in LMFDB)
3.9.af_t_abq$3$(not in LMFDB)
3.9.ac_ai_be$3$(not in LMFDB)
3.9.h_bc_dp$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.9.ag_l_am$2$(not in LMFDB)
3.9.ac_af_ci$2$(not in LMFDB)
3.9.c_af_aci$2$(not in LMFDB)
3.9.g_l_m$2$(not in LMFDB)
3.9.o_dn_nk$2$(not in LMFDB)
3.9.af_t_abq$3$(not in LMFDB)
3.9.ac_ai_be$3$(not in LMFDB)
3.9.h_bc_dp$3$(not in LMFDB)
3.9.ai_br_afo$4$(not in LMFDB)
3.9.ag_h_m$4$(not in LMFDB)
3.9.a_h_a$4$(not in LMFDB)
3.9.a_l_a$4$(not in LMFDB)
3.9.g_h_am$4$(not in LMFDB)
3.9.i_br_fo$4$(not in LMFDB)
3.9.al_cp_ajm$6$(not in LMFDB)
3.9.ak_bo_aek$6$(not in LMFDB)
3.9.ah_bc_adp$6$(not in LMFDB)
3.9.ad_l_ag$6$(not in LMFDB)
3.9.ab_e_abz$6$(not in LMFDB)
3.9.b_e_bz$6$(not in LMFDB)
3.9.c_ai_abe$6$(not in LMFDB)
3.9.d_l_g$6$(not in LMFDB)
3.9.f_t_bq$6$(not in LMFDB)
3.9.k_bo_ek$6$(not in LMFDB)
3.9.l_cp_jm$6$(not in LMFDB)
3.9.ae_q_acu$12$(not in LMFDB)
3.9.ad_h_g$12$(not in LMFDB)
3.9.d_h_ag$12$(not in LMFDB)
3.9.e_q_cu$12$(not in LMFDB)