Properties

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

Learn more about

Invariants

Base field:  $\F_{7}$
Dimension:  $3$
L-polynomial:  $( 1 - 5 x + 7 x^{2} )( 1 - 3 x + 7 x^{2} )( 1 - 2 x + 7 x^{2} )$
Frobenius angles:  $\pm0.106147807505$, $\pm0.308124534521$, $\pm0.376624142786$
Angle rank:  $3$ (numerical)

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

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
$A(\F_{q^r})$ 90 128700 46539360 14131260000 4677800643450 1617462425779200 558824103126708390 191865363956838000000 65755545242320535283360 22542577064915201368717500

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 54 394 2450 16558 116856 823954 5773346 40380118 282515814

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 1.7.ad $\times$ 1.7.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{7}$.

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.ag_u_acc$2$(not in LMFDB)
3.7.ae_k_aba$2$(not in LMFDB)
3.7.a_c_abe$2$(not in LMFDB)
3.7.a_c_be$2$(not in LMFDB)
3.7.e_k_ba$2$(not in LMFDB)
3.7.g_u_cc$2$(not in LMFDB)
3.7.k_ca_go$2$(not in LMFDB)
3.7.ae_w_aby$3$(not in LMFDB)
3.7.ab_h_k$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.ag_u_acc$2$(not in LMFDB)
3.7.ae_k_aba$2$(not in LMFDB)
3.7.a_c_abe$2$(not in LMFDB)
3.7.a_c_be$2$(not in LMFDB)
3.7.e_k_ba$2$(not in LMFDB)
3.7.g_u_cc$2$(not in LMFDB)
3.7.k_ca_go$2$(not in LMFDB)
3.7.ae_w_aby$3$(not in LMFDB)
3.7.ab_h_k$3$(not in LMFDB)
3.7.aj_bv_afu$6$(not in LMFDB)
3.7.ag_bg_adm$6$(not in LMFDB)
3.7.af_t_abu$6$(not in LMFDB)
3.7.ad_l_as$6$(not in LMFDB)
3.7.ac_q_aw$6$(not in LMFDB)
3.7.a_o_ag$6$(not in LMFDB)
3.7.a_o_g$6$(not in LMFDB)
3.7.b_h_ak$6$(not in LMFDB)
3.7.c_q_w$6$(not in LMFDB)
3.7.d_l_s$6$(not in LMFDB)
3.7.e_w_by$6$(not in LMFDB)
3.7.f_t_bu$6$(not in LMFDB)
3.7.g_bg_dm$6$(not in LMFDB)
3.7.j_bv_fu$6$(not in LMFDB)