Properties

Label 3.7.ak_bu_afk
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:  $( 1 + 7 x^{2} )( 1 - 5 x + 7 x^{2} )^{2}$
Frobenius angles:  $\pm0.106147807505$, $\pm0.106147807505$, $\pm0.5$
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
$A(\F_{q^r})$ 72 97344 36111744 13039812864 4762544934312 1645923601760256 560251539904401144 191705623808400000000 65753344523145944449152 22547288135426748620756544

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 42 304 2258 16858 118908 826054 5768546 40378768 282574842

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af 2 $\times$ 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 $\F_{7^{2}}$ is 1.49.al 2 $\times$ 1.49.o. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{7^{2}}$.

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.a_ae_a$2$(not in LMFDB)
3.7.k_bu_fk$2$(not in LMFDB)
3.7.ae_q_ace$3$(not in LMFDB)
3.7.ab_b_ao$3$(not in LMFDB)
3.7.c_w_bc$3$(not in LMFDB)
3.7.f_z_cs$3$(not in LMFDB)
3.7.i_bl_ei$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.a_ae_a$2$(not in LMFDB)
3.7.k_bu_fk$2$(not in LMFDB)
3.7.ae_q_ace$3$(not in LMFDB)
3.7.ab_b_ao$3$(not in LMFDB)
3.7.c_w_bc$3$(not in LMFDB)
3.7.f_z_cs$3$(not in LMFDB)
3.7.i_bl_ei$3$(not in LMFDB)
3.7.a_s_a$4$(not in LMFDB)
3.7.aj_bp_aew$6$(not in LMFDB)
3.7.ai_bl_aei$6$(not in LMFDB)
3.7.ag_ba_adg$6$(not in LMFDB)
3.7.af_z_acs$6$(not in LMFDB)
3.7.ad_r_abq$6$(not in LMFDB)
3.7.ac_w_abc$6$(not in LMFDB)
3.7.a_f_a$6$(not in LMFDB)
3.7.a_u_a$6$(not in LMFDB)
3.7.b_b_o$6$(not in LMFDB)
3.7.d_r_bq$6$(not in LMFDB)
3.7.e_q_ce$6$(not in LMFDB)
3.7.g_ba_dg$6$(not in LMFDB)
3.7.j_bp_ew$6$(not in LMFDB)
3.7.a_ag_a$12$(not in LMFDB)
3.7.a_j_a$12$(not in LMFDB)