Properties

Label 2.139.abo_zz
Base field $\F_{139}$
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:  $\F_{139}$
Dimension:  $2$
L-polynomial:  $1 - 40 x + 675 x^{2} - 5560 x^{3} + 19321 x^{4}$
Frobenius angles:  $\pm0.126846357897$, $\pm0.217883959392$
Angle rank:  $2$ (numerical)
Number field:  4.0.2679696.1
Galois group:  $D_{4}$
Jacobians:  $18$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14397$ $368520009$ $7213412355012$ $139367268840309081$ $2692484593399204872477$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $100$ $19072$ $2685940$ $373337476$ $51889468900$ $7212556397662$ $1002544422864700$ $139353667441169668$ $19370159741633281900$ $2692452204178120224352$

Jacobians and polarizations

This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{139}$.

Endomorphism algebra over $\F_{139}$
The endomorphism algebra of this simple isogeny class is 4.0.2679696.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.139.bo_zz$2$(not in LMFDB)