Properties

Label 2.137.abo_zs
Base field $\F_{137}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{137}$
Dimension:  $2$
L-polynomial:  $1 - 40 x + 668 x^{2} - 5480 x^{3} + 18769 x^{4}$
Frobenius angles:  $\pm0.0914703999019$, $\pm0.230187862161$
Angle rank:  $2$ (numerical)
Number field:  4.0.6084864.2
Galois group:  $D_{4}$
Jacobians:  $12$

This isogeny class is simple and geometrically 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $13918$ $347365444$ $6611137641646$ $124105337760005776$ $2329210080181838052718$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $98$ $18506$ $2571074$ $352296390$ $48262056658$ $6611858912522$ $905824311389074$ $124097929817530174$ $17001416404541246498$ $2329194047600197425386$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 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 endomorphism algebra of this simple isogeny class is 4.0.6084864.2.

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.bo_zs$2$(not in LMFDB)