Properties

Label 2.139.abo_zw
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

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{139}$
Dimension:  $2$
L-polynomial:  $1 - 40 x + 672 x^{2} - 5560 x^{3} + 19321 x^{4}$
Frobenius angles:  $\pm0.0989502623149$, $\pm0.232779741831$
Angle rank:  $2$ (numerical)
Number field:  4.0.7430400.5
Galois group:  $D_{4}$
Jacobians:  $32$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14394$ $368400036$ $7212443624874$ $139363117978544400$ $2692472389058327844954$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $100$ $19066$ $2685580$ $373326358$ $51889233700$ $7212552684586$ $1002544379408140$ $139353667126146718$ $19370159742298074820$ $2692452204247217689306$

Jacobians and polarizations

This isogeny class contains the Jacobians of 32 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 4.0.7430400.5.

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_zw$2$(not in LMFDB)