Properties

Label 2.137.abn_zc
Base field $\F_{137}$
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:  $\F_{137}$
Dimension:  $2$
L-polynomial:  $( 1 - 21 x + 137 x^{2} )( 1 - 18 x + 137 x^{2} )$
  $1 - 39 x + 652 x^{2} - 5343 x^{3} + 18769 x^{4}$
Frobenius angles:  $\pm0.145687313345$, $\pm0.220793476886$
Angle rank:  $2$ (numerical)
Jacobians:  $36$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14040$ $348248160$ $6614267194080$ $124113673139414400$ $2329228076205097618200$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $99$ $18553$ $2572290$ $352320049$ $48262429539$ $6611863686766$ $905824357825563$ $124097930056017121$ $17001416401942187250$ $2329194047499778932793$

Jacobians and polarizations

This isogeny class contains the Jacobians of 36 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{137}$.

Endomorphism algebra over $\F_{137}$
The isogeny class factors as 1.137.av $\times$ 1.137.as 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
2.137.ad_aea$2$(not in LMFDB)
2.137.d_aea$2$(not in LMFDB)
2.137.bn_zc$2$(not in LMFDB)