Abelian variety isogeny class 2.137.abn_zb over $\F_{137}$

• Label: 2.137.abn_zb
• Base field: $\F_{137}$
• 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:	$\F_{137}$
Dimension:	$2$
L-polynomial:	$1 - 39 x + 651 x^{2} - 5343 x^{3} + 18769 x^{4}$
Frobenius angles:	$\pm0.136074106037$, $\pm0.227156831293$
Angle rank:	$2$ (numerical)
Number field:	4.0.3737773.2
Galois group:	$D_{4}$
Jacobians:	$10$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14039$	$348209317$	$6613965753743$	$124112447775085909$	$2329224735261325160624$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$99$	$18551$	$2572173$	$352316571$	$48262360314$	$6611862708263$	$905824349047521$	$124097930053524499$	$17001416403682114959$	$2329194047540811591566$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+120x^5+13x^4+10x^3+42x^2+103x+86$
• $y^2=64x^6+14x^5+10x^4+12x^3+86x^2+81x+85$
• $y^2=94x^6+112x^5+83x^4+115x^3+112x^2+23x+40$
• $y^2=6x^6+28x^5+124x^4+44x^2+60x+6$
• $y^2=104x^6+105x^5+76x^4+38x^3+47x^2+131x+99$
• $y^2=42x^6+90x^5+70x^4+63x^3+40x^2+123x+67$
• $y^2=35x^6+108x^5+79x^4+14x^3+40x^2+84$
• $y^2=8x^6+52x^5+73x^4+85x^3+56x^2+34x+15$
• $y^2=86x^6+90x^5+61x^4+14x^3+56x^2+49x+10$
• $y^2=126x^6+74x^5+124x^4+46x^3+36x^2+91x+76$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{137}$.

Endomorphism algebra over $\F_{137}$

The endomorphism algebra of this simple isogeny class is 4.0.3737773.2.

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
2.137.bn_zb	$2$	(not in LMFDB)