Abelian variety isogeny class 2.139.abo_zx over $\F_{139}$

• Label: 2.139.abo_zx
• Base field: $\F_{139}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{139}$
Dimension:	$2$
L-polynomial:	$1 - 40 x + 673 x^{2} - 5560 x^{3} + 19321 x^{4}$
Frobenius angles:	$\pm0.107972453136$, $\pm0.228432515374$
Angle rank:	$2$ (numerical)
Number field:	4.0.1025.1
Galois group:	$D_{4}$
Jacobians:	$38$

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14395$	$368440025$	$7212766530880$	$139364503088830025$	$2692476477925650389875$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$100$	$19068$	$2685700$	$373330068$	$51889312500$	$7212553942998$	$1002544394632300$	$139353667251377508$	$19370159742524743900$	$2692452204232422219148$

Jacobians and polarizations

This isogeny class contains the Jacobians of 38 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=84x^6+99x^5+54x^4+17x^3+27x^2+39x+93$
• $y^2=37x^6+129x^5+72x^4+5x^3+53x^2+33x+110$
• $y^2=86x^6+135x^5+22x^4+88x^3+37x^2+17x+100$
• $y^2=134x^6+111x^5+105x^4+11x^3+98x^2+69x+53$
• $y^2=14x^6+125x^5+107x^4+95x^3+72x^2+71x+62$
• $y^2=42x^6+134x^5+26x^4+118x^3+52x^2+119x+8$
• $y^2=66x^6+129x^5+42x^4+102x^3+62x^2+94x+98$
• $y^2=18x^6+132x^5+53x^4+51x^3+83x^2+96x+17$
• $y^2=87x^6+8x^5+60x^4+60x^3+x^2+89x+57$
• $y^2=61x^6+101x^5+119x^4+12x^3+100x^2+90x+28$
• $y^2=14x^6+59x^5+111x^4+63x^3+11x^2+111x+134$
• $y^2=99x^6+93x^5+69x^4+71x^3+7x^2+3x+96$
• $y^2=79x^6+130x^5+102x^4+121x^3+33x^2+8x+82$
• $y^2=133x^6+39x^5+80x^4+10x^3+113x^2+67x+51$
• $y^2=80x^6+109x^5+123x^4+96x^3+123x^2+92x+53$
• $y^2=15x^6+64x^5+46x^4+112x^3+138x^2+87x+108$
• $y^2=6x^6+103x^5+86x^4+52x^3+76x^2+37x+114$
• $y^2=104x^6+123x^5+62x^4+60x^3+65x^2+84x+20$
• $y^2=37x^6+41x^5+90x^4+x^3+114x^2+67x+26$
• $y^2=110x^6+76x^5+11x^4+101x^3+83x^2+79x+13$
• and

• $y^2=12x^6+138x^5+93x^4+76x^3+41x^2+50x+22$
• $y^2=79x^6+53x^5+14x^4+119x^3+73x^2+100x+82$
• $y^2=41x^6+111x^5+123x^4+37x^3+52x^2+90x+73$
• $y^2=47x^6+106x^5+20x^4+78x^3+129x^2+12x+127$
• $y^2=64x^6+82x^5+63x^3+83x^2+48x+104$
• $y^2=55x^6+88x^5+127x^4+22x^3+31x^2+49x+47$
• $y^2=120x^6+76x^5+70x^4+57x^3+125x^2+36x+26$
• $y^2=23x^6+39x^5+106x^4+32x^3+60x^2+125x+35$
• $y^2=22x^6+86x^5+3x^4+25x^3+134x^2+5x+23$
• $y^2=6x^6+39x^5+38x^4+132x^3+2x^2+108x+53$
• $y^2=52x^6+74x^5+110x^4+27x^3+15x^2+138x+134$
• $y^2=11x^6+67x^5+23x^4+25x^3+27x^2+79x+85$
• $y^2=29x^6+99x^5+51x^4+5x^3+54x^2+87x+135$
• $y^2=52x^6+62x^5+99x^4+47x^3+6x^2+55x+29$
• $y^2=16x^6+99x^5+103x^4+37x^3+115x^2+111x+104$
• $y^2=110x^6+34x^5+123x^4+42x^3+22x^2+20x+43$
• $y^2=25x^6+135x^5+15x^4+115x^3+31x^2+71x+122$
• $y^2=114x^6+126x^5+35x^4+x^3+14x^2+41x+108$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{139}$.

Endomorphism algebra over $\F_{139}$

The endomorphism algebra of this simple isogeny class is 4.0.1025.1.

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_zx	$2$	(not in LMFDB)