Properties

Label 2.83.abf_pk
Base field $\F_{83}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{83}$
Dimension:  $2$
L-polynomial:  $( 1 - 18 x + 83 x^{2} )( 1 - 13 x + 83 x^{2} )$
  $1 - 31 x + 400 x^{2} - 2573 x^{3} + 6889 x^{4}$
Frobenius angles:  $\pm0.0496118990883$, $\pm0.247123549255$
Angle rank:  $2$ (numerical)
Jacobians:  $3$
Isomorphism classes:  13

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4686$ $46363284$ $326762858664$ $2252414757881376$ $15516054496826400066$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $53$ $6729$ $571478$ $47460905$ $3939044023$ $326939756418$ $27136039294165$ $2252292107260369$ $186940254463791314$ $15516041186398856889$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{83}$.

Endomorphism algebra over $\F_{83}$
The isogeny class factors as 1.83.as $\times$ 1.83.an 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 is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.83.af_acq$2$(not in LMFDB)
2.83.f_acq$2$(not in LMFDB)
2.83.bf_pk$2$(not in LMFDB)