# Properties

 Label 2.139.abr_bcj 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: $1 - 43 x + 737 x^{2} - 5977 x^{3} + 19321 x^{4}$ Frobenius angles: $\pm0.0488276070962$, $\pm0.185265935553$ Angle rank: $2$ (numerical) Number field: 4.0.368589.1 Galois group: $D_{4}$ Jacobians: 6

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 6 curves, and hence is principally polarizable:

• $y^2=74x^6+16x^5+42x^4+133x^3+86x^2+100x+40$
• $y^2=138x^6+136x^5+81x^4+78x^3+94x^2+33x+19$
• $y^2=93x^6+97x^5+35x^4+34x^3+70x^2+54x+126$
• $y^2=18x^6+77x^5+101x^4+x^3+29x^2+87x+32$
• $y^2=81x^6+77x^5+121x^4+51x^3+7x^2+4x+115$
• $y^2=61x^6+124x^5+38x^4+73x^3+102x^2+96x+19$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 14039 366123081 7206201614537 139351783069244061 2692458192750605660624 52020881963077220947780569 1005095218219601866673843227229 19419444536131486282313517718878549 375203088329197597078858623562025599307 7249298871624660761746683862690195859602176

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 97 18947 2683255 373295995 51888960112 7212551205287 1002544375960357 139353667002050179 19370159736341116855 2692452204100261958102

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{139}$
 The endomorphism algebra of this simple isogeny class is 4.0.368589.1.
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_bcj $2$ (not in LMFDB)