Properties

Label 3.7.aj_bp_aew
Base Field $\F_{7}$
Dimension $3$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{7}$
Dimension:  $3$
L-polynomial:  $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 7 x^{2} )( 1 + 7 x^{2} )$
Frobenius angles:  $\pm0.106147807505$, $\pm0.227185525829$, $\pm0.5$
Angle rank:  $1$ (numerical)

This isogeny class is not simple.

Newton polygon

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

Point counts

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

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 96 119808 40569984 13681115136 4822242966816 1645923601760256 559054007578523616 191429583261696000000 65712360862558465840512 22542892859348070835811328

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 51 344 2375 17069 118908 824291 5760239 40353608 282519771

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 1.7.ae $\times$ 1.7.a 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 $\overline{\F}_{7}$
The base change of $A$ to $\F_{7^{6}}$ is 1.117649.la 2 $\times$ 1.117649.bak. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{7^{6}}$.
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
3.7.ab_b_ao$2$(not in LMFDB)
3.7.b_b_o$2$(not in LMFDB)
3.7.ag_ba_adg$3$(not in LMFDB)
3.7.ad_r_abq$3$(not in LMFDB)
3.7.a_ae_a$3$(not in LMFDB)
3.7.a_f_a$3$(not in LMFDB)
3.7.a_u_a$3$(not in LMFDB)
3.7.d_r_bq$3$(not in LMFDB)
3.7.g_ba_dg$3$(not in LMFDB)
3.7.j_bp_ew$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.ab_b_ao$2$(not in LMFDB)
3.7.b_b_o$2$(not in LMFDB)
3.7.ag_ba_adg$3$(not in LMFDB)
3.7.ad_r_abq$3$(not in LMFDB)
3.7.a_ae_a$3$(not in LMFDB)
3.7.a_f_a$3$(not in LMFDB)
3.7.a_u_a$3$(not in LMFDB)
3.7.d_r_bq$3$(not in LMFDB)
3.7.g_ba_dg$3$(not in LMFDB)
3.7.j_bp_ew$3$(not in LMFDB)
3.7.ak_bu_afk$6$(not in LMFDB)
3.7.ai_bl_aei$6$(not in LMFDB)
3.7.af_z_acs$6$(not in LMFDB)
3.7.ae_q_ace$6$(not in LMFDB)
3.7.ac_w_abc$6$(not in LMFDB)
3.7.c_w_bc$6$(not in LMFDB)
3.7.d_r_bq$6$(not in LMFDB)
3.7.e_q_ce$6$(not in LMFDB)
3.7.f_z_cs$6$(not in LMFDB)
3.7.i_bl_ei$6$(not in LMFDB)
3.7.k_bu_fk$6$(not in LMFDB)
3.7.a_ag_a$12$(not in LMFDB)
3.7.a_j_a$12$(not in LMFDB)
3.7.a_s_a$12$(not in LMFDB)