# Properties

 Label 1.139.c 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: $1 + 2 x + 139 x^{2}$ Frobenius angles: $\pm0.527031169954$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-138})$$ 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:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 142 19596 2684794 373264608 51889032382 7212554102124 1002544332963658 139353666630849408 19370159748515435566 2692452204265493556876

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 142 19596 2684794 373264608 51889032382 7212554102124 1002544332963658 139353666630849408 19370159748515435566 2692452204265493556876

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{139}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-138})$$.
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.
 Twist Extension Degree Common base change 1.139.ac $2$ (not in LMFDB)