Properties

Label 3.7.aj_bo_aer
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

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{7}$
Dimension:  $3$
L-polynomial:  $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 13 x^{2} - 28 x^{3} + 49 x^{4} )$
  $1 - 9 x + 40 x^{2} - 121 x^{3} + 280 x^{4} - 441 x^{5} + 343 x^{6}$
Frobenius angles:  $\pm0.106147807505$, $\pm0.204545263622$, $\pm0.514205363720$
Angle rank:  $3$ (numerical)
Isomorphism classes:  24

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $93$ $114855$ $39211776$ $13626971475$ $4845127076823$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $49$ $332$ $2365$ $17149$ $119116$ $824627$ $5762581$ $40360004$ $282505289$

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 $\times$ 2.7.ae_n 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.ab_a_aj$2$(not in LMFDB)
3.7.b_a_j$2$(not in LMFDB)
3.7.j_bo_er$2$(not in LMFDB)
3.7.ad_q_abr$3$(not in LMFDB)
3.7.a_e_ae$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.7.ab_a_aj$2$(not in LMFDB)
3.7.b_a_j$2$(not in LMFDB)
3.7.j_bo_er$2$(not in LMFDB)
3.7.ad_q_abr$3$(not in LMFDB)
3.7.a_e_ae$3$(not in LMFDB)
3.7.ai_bk_aee$6$(not in LMFDB)
3.7.af_y_acr$6$(not in LMFDB)
3.7.a_e_e$6$(not in LMFDB)
3.7.d_q_br$6$(not in LMFDB)
3.7.f_y_cr$6$(not in LMFDB)
3.7.i_bk_ee$6$(not in LMFDB)