# Properties

 Label 2.139.abo_zw 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: $1 - 40 x + 672 x^{2} - 5560 x^{3} + 19321 x^{4}$ Frobenius angles: $\pm0.0989502623149$, $\pm0.232779741831$ Angle rank: $2$ (numerical) Number field: 4.0.7430400.5 Galois group: $D_{4}$ Jacobians: 32

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 32 curves, and hence is principally polarizable:

• $y^2=72x^6+111x^5+11x^4+34x^3+132x^2+46x+4$
• $y^2=20x^6+70x^5+24x^4+118x^3+47x^2+79x+25$
• $y^2=102x^6+40x^5+25x^4+26x^3+138x^2+120x+99$
• $y^2=35x^6+133x^5+46x^4+4x^3+53x^2+52x+41$
• $y^2=39x^6+100x^5+18x^4+120x^3+48x^2+109x+56$
• $y^2=10x^6+101x^5+131x^4+43x^3+47x^2+50x+138$
• $y^2=59x^6+59x^5+25x^4+137x^3+119x^2+53x+126$
• $y^2=39x^6+75x^5+65x^4+51x^3+109x+127$
• $y^2=86x^6+64x^5+31x^4+14x^3+10x^2+43x+82$
• $y^2=59x^6+126x^5+52x^4+81x^3+118x^2+50x+102$
• $y^2=9x^6+74x^5+113x^4+110x^3+13x^2+39x+11$
• $y^2=80x^6+93x^5+117x^4+134x^3+100x^2+3x+48$
• $y^2=34x^6+62x^5+132x^4+42x^3+71x^2+7x+49$
• $y^2=35x^6+14x^5+100x^4+32x^3+92x^2+58x+12$
• $y^2=7x^6+22x^5+111x^4+95x^3+117x^2+81x+61$
• $y^2=26x^6+94x^5+125x^4+115x^3+72x^2+68x+97$
• $y^2=48x^6+81x^5+19x^4+54x^3+118x^2+60x+88$
• $y^2=103x^6+28x^5+100x^4+42x^3+37x^2+7x+27$
• $y^2=77x^6+106x^5+108x^4+71x^3+101x^2+106x+60$
• $y^2=95x^6+96x^5+111x^4+15x^3+125x^2+27x+38$
• and 12 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 14394 368400036 7212443624874 139363117978544400 2692472389058327844954 52020892632613431560466276 1005095221676157533415778639434 19419444553424793689389820348006400 375203088444584824362740134926086795514 7249298872020332044140368631764524986862116

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 100 19066 2685580 373326358 51889233700 7212552684586 1002544379408140 139353667126146718 19370159742298074820 2692452204247217689306

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{139}$
 The endomorphism algebra of this simple isogeny class is 4.0.7430400.5.
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.bo_zw $2$ (not in LMFDB)