Properties

Label 3.9.an_cx_akk
Base field $\F_{3^{2}}$
Dimension $3$
$p$-rank $1$
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:  $( 1 - 3 x )^{4}( 1 - x + 9 x^{2} )$
  $1 - 13 x + 75 x^{2} - 270 x^{3} + 675 x^{4} - 1053 x^{5} + 729 x^{6}$
Frobenius angles:  $0$, $0$, $0$, $0$, $\pm0.446699620962$
Angle rank:  $1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $144$ $405504$ $345473856$ $263577600000$ $201288133872144$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-3$ $63$ $648$ $6111$ $57717$ $529308$ $4778253$ $43017471$ $387302472$ $3486535983$

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 2 $\times$ 1.9.ab 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.al_bz_agg$2$(not in LMFDB)
3.9.ab_aj_s$2$(not in LMFDB)
3.9.b_aj_as$2$(not in LMFDB)
3.9.l_bz_gg$2$(not in LMFDB)
3.9.n_cx_kk$2$(not in LMFDB)
3.9.ae_m_acc$3$(not in LMFDB)
3.9.f_be_dd$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.9.al_bz_agg$2$(not in LMFDB)
3.9.ab_aj_s$2$(not in LMFDB)
3.9.b_aj_as$2$(not in LMFDB)
3.9.l_bz_gg$2$(not in LMFDB)
3.9.n_cx_kk$2$(not in LMFDB)
3.9.ae_m_acc$3$(not in LMFDB)
3.9.f_be_dd$3$(not in LMFDB)
3.9.ah_bh_aew$4$(not in LMFDB)
3.9.af_v_adm$4$(not in LMFDB)
3.9.ab_bb_as$4$(not in LMFDB)
3.9.b_bb_s$4$(not in LMFDB)
3.9.f_v_dm$4$(not in LMFDB)
3.9.h_bh_ew$4$(not in LMFDB)
3.9.c_p_bt$5$(not in LMFDB)
3.9.ak_cc_ahq$6$(not in LMFDB)
3.9.ai_bk_aew$6$(not in LMFDB)
3.9.ah_bq_aff$6$(not in LMFDB)
3.9.af_be_add$6$(not in LMFDB)
3.9.ac_g_acc$6$(not in LMFDB)
3.9.ab_s_aj$6$(not in LMFDB)
3.9.b_s_j$6$(not in LMFDB)
3.9.c_g_cc$6$(not in LMFDB)
3.9.e_m_cc$6$(not in LMFDB)
3.9.h_bq_ff$6$(not in LMFDB)
3.9.i_bk_ew$6$(not in LMFDB)
3.9.k_cc_hq$6$(not in LMFDB)
3.9.ab_j_a$8$(not in LMFDB)
3.9.b_j_a$8$(not in LMFDB)
3.9.ae_v_acl$10$(not in LMFDB)
3.9.ac_p_abt$10$(not in LMFDB)
3.9.e_v_cl$10$(not in LMFDB)
3.9.ae_be_acu$12$(not in LMFDB)
3.9.ac_y_abk$12$(not in LMFDB)
3.9.ab_a_j$12$(not in LMFDB)
3.9.b_a_aj$12$(not in LMFDB)
3.9.c_y_bk$12$(not in LMFDB)
3.9.e_be_cu$12$(not in LMFDB)