Properties

Label 1.139.ai
Base Field $\F_{139}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{139}$
Dimension:  $1$
L-polynomial:  $1 - 8 x + 139 x^{2}$
Frobenius angles:  $\pm0.389818088061$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-123}) \)
Galois group:  $C_2$
Jacobians:  8

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

This isogeny class contains the Jacobians of 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 132 19536 2688444 373293888 51888394932 7212546809424 1002544410117228 139353667907106048 19370159742192799716 2692452204098426836176

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 132 19536 2688444 373293888 51888394932 7212546809424 1002544410117228 139353667907106048 19370159742192799716 2692452204098426836176

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{139}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-123}) \).
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
1.139.i$2$(not in LMFDB)