Properties

Label 2.89.ae_bm
Base field $\F_{89}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{89}$
Dimension:  $2$
L-polynomial:  $( 1 - 14 x + 89 x^{2} )( 1 + 10 x + 89 x^{2} )$
  $1 - 4 x + 38 x^{2} - 356 x^{3} + 7921 x^{4}$
Frobenius angles:  $\pm0.233878122877$, $\pm0.677807684489$
Angle rank:  $2$ (numerical)
Jacobians:  $952$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $7600$ $63232000$ $496504481200$ $3938174955520000$ $31182724162624078000$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $86$ $7982$ $704294$ $62767518$ $5584239286$ $496980333902$ $44231338586374$ $3936588720644158$ $350356401461807126$ $31181719933110485102$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 952 curves (of which all are hyperelliptic):

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{89}$.

Endomorphism algebra over $\F_{89}$
The isogeny class factors as 1.89.ao $\times$ 1.89.k 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
2.89.ay_mg$2$(not in LMFDB)
2.89.e_bm$2$(not in LMFDB)
2.89.y_mg$2$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.89.ay_mg$2$(not in LMFDB)
2.89.e_bm$2$(not in LMFDB)
2.89.y_mg$2$(not in LMFDB)
2.89.abe_pm$4$(not in LMFDB)
2.89.ac_abu$4$(not in LMFDB)
2.89.c_abu$4$(not in LMFDB)
2.89.be_pm$4$(not in LMFDB)