Properties

Label 2.139.abn_za
Base Field $\F_{139}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{139}$
Dimension:  $2$
L-polynomial:  $1 - 39 x + 650 x^{2} - 5421 x^{3} + 19321 x^{4}$
Frobenius angles:  $\pm0.102302809608$, $\pm0.250869615261$
Angle rank:  $2$ (numerical)
Number field:  4.0.4222053.1
Galois group:  $D_{4}$
Jacobians:  44

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 44 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 14512 369069184 7213807288768 139364032988201472 2692469545090851881872 52020881961239836934232064 1005095204795937125971813008976 19419444548895775329173132215584768 375203088494703620514580679876415004096 7249298872171281075906944099480173718675584

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 101 19101 2686088 373328809 51889178891 7212551205030 1002544362570761 139353667093646545 19370159744885497784 2692452204303281463261

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{139}$
The endomorphism algebra of this simple isogeny class is 4.0.4222053.1.
All geometric endomorphisms are defined over $\F_{139}$.

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.bn_za$2$(not in LMFDB)