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

• Label: 2.139.abn_zb
• 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:	$1 - 39 x + 651 x^{2} - 5421 x^{3} + 19321 x^{4}$
Frobenius angles:	$\pm0.109723050661$, $\pm0.247429772818$
Angle rank:	$2$ (numerical)
Number field:	4.0.14603965.2
Galois group:	$D_{4}$
Jacobians:	$28$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14513$	$369109129$	$7214122083539$	$139365332957794285$	$2692473177587401560848$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$101$	$19103$	$2686205$	$373332291$	$51889248896$	$7212552236291$	$1002544373497859$	$139353667158881971$	$19370159744559896495$	$2692452204287543592278$

Jacobians and polarizations

This isogeny class contains the Jacobians of 28 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=112x^5+67x^4+66x^3+5x^2+77x+52$
• $y^2=73x^6+21x^5+25x^4+18x^3+75x^2+34x+92$
• $y^2=7x^6+138x^5+126x^4+35x^3+108x^2+88x+8$
• $y^2=110x^6+132x^5+66x^4+43x^3+55x^2+28x+122$
• $y^2=130x^6+92x^5+20x^4+63x^3+38x^2+134x+12$
• $y^2=41x^6+49x^5+44x^4+6x^3+122x^2+100x+111$
• $y^2=9x^6+59x^5+19x^4+46x^3+101x^2+128x+119$
• $y^2=92x^6+73x^5+15x^4+10x^3+77x^2+125x+41$
• $y^2=22x^6+108x^5+119x^4+18x^3+34x^2+48x+21$
• $y^2=123x^6+121x^5+129x^4+58x^3+13x^2+99x+131$
• $y^2=22x^6+113x^5+60x^4+48x^3+x^2+127x+51$
• $y^2=82x^6+73x^5+118x^4+4x^3+46x^2+17x+82$
• $y^2=50x^6+83x^5+7x^4+109x^3+121x^2+76x+109$
• $y^2=123x^6+79x^5+33x^4+86x^3+30x^2+94x+4$
• $y^2=76x^6+23x^5+65x^4+76x^3+42x^2+6x+53$
• $y^2=53x^6+83x^5+81x^4+7x^3+121x^2+33x+53$
• $y^2=85x^6+86x^5+30x^4+105x^3+86x^2+126x+136$
• $y^2=115x^6+107x^5+15x^4+97x^3+21x^2+26x+58$
• $y^2=54x^6+71x^5+16x^4+38x^3+10x^2+125x+106$
• $y^2=13x^6+55x^5+88x^4+25x^3+61x^2+33x+108$
• $y^2=45x^6+29x^5+111x^4+127x^3+90x^2+109x+60$
• $y^2=106x^6+65x^5+86x^4+48x^3+111x^2+107x+135$
• $y^2=46x^6+79x^5+66x^4+135x^3+61x^2+87x+45$
• $y^2=115x^6+28x^5+45x^4+130x^3+138x^2+76x+80$
• $y^2=92x^6+60x^5+101x^4+56x^3+105x^2+117x+58$
• $y^2=52x^6+94x^5+7x^4+15x^3+53x^2+49x+115$
• $y^2=26x^6+17x^5+44x^4+88x^3+72x^2+70x+98$
• $y^2=102x^6+93x^5+134x^4+77x^3+80x^2+55x+138$

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 4.0.14603965.2.

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.bn_zb	$2$	(not in LMFDB)