Properties

Label 3.7.an_cy_ajx
Base field $\F_{7}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{7}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x + 7 x^{2} )( 1 - 5 x + 7 x^{2} )^{2}$
  $1 - 13 x + 76 x^{2} - 257 x^{3} + 532 x^{4} - 637 x^{5} + 343 x^{6}$
Frobenius angles:  $\pm0.106147807505$, $\pm0.106147807505$, $\pm0.308124534521$
Angle rank:  $2$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $45$ $83655$ $39890880$ $14007611475$ $4753194387975$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-5$ $33$ $340$ $2429$ $16825$ $117612$ $824455$ $5772821$ $40388380$ $282573753$

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 $\F_{7}$.

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af 2 $\times$ 1.7.ad 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
3.7.ah_q_ax$2$(not in LMFDB)
3.7.ad_ae_bh$2$(not in LMFDB)
3.7.d_ae_abh$2$(not in LMFDB)
3.7.h_q_x$2$(not in LMFDB)
3.7.n_cy_jx$2$(not in LMFDB)
3.7.ah_bc_adf$3$(not in LMFDB)
3.7.ae_e_e$3$(not in LMFDB)
3.7.ab_q_ar$3$(not in LMFDB)
3.7.c_k_q$3$(not in LMFDB)
3.7.f_n_w$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.7.ah_q_ax$2$(not in LMFDB)
3.7.ad_ae_bh$2$(not in LMFDB)
3.7.d_ae_abh$2$(not in LMFDB)
3.7.h_q_x$2$(not in LMFDB)
3.7.n_cy_jx$2$(not in LMFDB)
3.7.ah_bc_adf$3$(not in LMFDB)
3.7.ae_e_e$3$(not in LMFDB)
3.7.ab_q_ar$3$(not in LMFDB)
3.7.c_k_q$3$(not in LMFDB)
3.7.f_n_w$3$(not in LMFDB)
3.7.ad_s_abh$4$(not in LMFDB)
3.7.d_s_bh$4$(not in LMFDB)
3.7.am_cq_aiu$6$(not in LMFDB)
3.7.al_cj_ahu$6$(not in LMFDB)
3.7.aj_bs_afl$6$(not in LMFDB)
3.7.ai_bo_aeu$6$(not in LMFDB)
3.7.ag_o_ay$6$(not in LMFDB)
3.7.ag_ba_acu$6$(not in LMFDB)
3.7.af_n_aw$6$(not in LMFDB)
3.7.af_bc_acv$6$(not in LMFDB)
3.7.ad_f_g$6$(not in LMFDB)
3.7.ad_i_abb$6$(not in LMFDB)
3.7.ad_u_abn$6$(not in LMFDB)
3.7.ac_ac_bg$6$(not in LMFDB)
3.7.ac_k_aq$6$(not in LMFDB)
3.7.ab_e_abd$6$(not in LMFDB)
3.7.a_i_am$6$(not in LMFDB)
3.7.a_i_m$6$(not in LMFDB)
3.7.b_e_bd$6$(not in LMFDB)
3.7.b_q_r$6$(not in LMFDB)
3.7.c_ac_abg$6$(not in LMFDB)
3.7.d_f_ag$6$(not in LMFDB)
3.7.d_i_bb$6$(not in LMFDB)
3.7.d_u_bn$6$(not in LMFDB)
3.7.e_e_ae$6$(not in LMFDB)
3.7.f_bc_cv$6$(not in LMFDB)
3.7.g_o_y$6$(not in LMFDB)
3.7.g_ba_cu$6$(not in LMFDB)
3.7.h_bc_df$6$(not in LMFDB)
3.7.i_bo_eu$6$(not in LMFDB)
3.7.j_bs_fl$6$(not in LMFDB)
3.7.l_cj_hu$6$(not in LMFDB)
3.7.m_cq_iu$6$(not in LMFDB)
3.7.ad_ag_bn$12$(not in LMFDB)
3.7.ad_j_ag$12$(not in LMFDB)
3.7.d_ag_abn$12$(not in LMFDB)
3.7.d_j_g$12$(not in LMFDB)