# Properties

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

## Invariants

 Base field: $\F_{139}$ Dimension: $2$ L-polynomial: $1 - 43 x + 736 x^{2} - 5977 x^{3} + 19321 x^{4}$ Frobenius angles: $\pm0.0124725911818$, $\pm0.191527477370$ Angle rank: $2$ (numerical) Number field: 4.0.43928.1 Galois group: $D_{4}$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

• $y^2=43x^6+100x^5+110x^4+115x^3+71x^2+59x+21$
• $y^2=61x^6+51x^5+113x^4+41x^3+99x^2+76x+29$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 14038 366082964 7205854371256 139350121623952928 2692452447301948800538 52020865926592467382152896 1005095180019665113699037088682 19419444455859695708079116940211328 375203088176794147436563335970331350552 7249298871357146940504408755803444011913044

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 97 18945 2683126 373291545 51888849387 7212548981874 1002544337857369 139353666426020913 19370159728473166786 2692452204000905014705

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{139}$
 The endomorphism algebra of this simple isogeny class is 4.0.43928.1.
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.
 Twist Extension Degree Common base change 2.139.br_bci $2$ (not in LMFDB)