Properties

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

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Invariants

Base field:  $\F_{7}$
Dimension:  $3$
L-polynomial:  $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 8 x^{2} - 28 x^{3} + 49 x^{4} )$
Frobenius angles:  $\pm0.0704914820143$, $\pm0.106147807505$, $\pm0.570491482014$
Angle rank:  $2$ (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
$A(\F_{q^r})$ 78 91260 32760936 13026452400 4787838907098 1632410608570560 558141140442602886 191787062249502720000 65760214491382195402488 22541972682167303266921500

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 39 272 2255 16949 117936 822947 5770991 40382984 282508239

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 2.7.ae_i 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 $\F_{7^{4}}$ is 1.2401.ack 2 $\times$ 1.2401.ax. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{7^{4}}$.
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_af_q$2$(not in LMFDB)
3.7.b_af_aq$2$(not in LMFDB)
3.7.j_bj_ds$2$(not in LMFDB)
3.7.ad_l_abw$3$(not in LMFDB)
3.7.a_ab_ay$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.ab_af_q$2$(not in LMFDB)
3.7.b_af_aq$2$(not in LMFDB)
3.7.j_bj_ds$2$(not in LMFDB)
3.7.ad_l_abw$3$(not in LMFDB)
3.7.a_ab_ay$3$(not in LMFDB)
3.7.ai_bf_adk$6$(not in LMFDB)
3.7.af_t_acm$6$(not in LMFDB)
3.7.a_ab_y$6$(not in LMFDB)
3.7.d_l_bw$6$(not in LMFDB)
3.7.f_t_cm$6$(not in LMFDB)
3.7.i_bf_dk$6$(not in LMFDB)
3.7.af_b_be$8$(not in LMFDB)
3.7.af_n_abe$8$(not in LMFDB)
3.7.f_b_abe$8$(not in LMFDB)
3.7.f_n_be$8$(not in LMFDB)
3.7.ae_b_y$24$(not in LMFDB)
3.7.ae_n_ay$24$(not in LMFDB)
3.7.ab_b_g$24$(not in LMFDB)
3.7.ab_n_ag$24$(not in LMFDB)
3.7.b_b_ag$24$(not in LMFDB)
3.7.b_n_g$24$(not in LMFDB)
3.7.e_b_ay$24$(not in LMFDB)
3.7.e_n_y$24$(not in LMFDB)