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

• Label: 2.137.abo_zr
• 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:	$1 - 40 x + 667 x^{2} - 5480 x^{3} + 18769 x^{4}$
Frobenius angles:	$\pm0.0815275655821$, $\pm0.234192154975$
Angle rank:	$2$ (numerical)
Number field:	4.0.6805904.1
Galois group:	$D_{4}$
Jacobians:	$14$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13917$	$347326569$	$6610828456404$	$124104023613780921$	$2329206199914677989557$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$18504$	$2570954$	$352292660$	$48261976258$	$6611857585398$	$905824294074994$	$124097929642236964$	$17001416403319453178$	$2329194047598123745464$

Jacobians and polarizations

This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=55x^6+48x^5+135x^4+45x^3+127x^2+103x+72$
• $y^2=119x^6+15x^5+102x^4+13x^3+56x^2+124x+31$
• $y^2=127x^6+12x^5+134x^4+112x^3+76x^2+32x+4$
• $y^2=39x^6+121x^5+85x^4+111x^3+114x^2+125x+11$
• $y^2=18x^6+32x^5+29x^4+52x^3+25x^2+43x+127$
• $y^2=35x^6+133x^5+10x^4+82x^3+74x^2+23x+96$
• $y^2=20x^6+59x^5+111x^4+50x^3+2x^2+48x+26$
• $y^2=105x^6+104x^5+88x^4+17x^3+29x^2+23x+112$
• $y^2=55x^6+130x^5+131x^4+109x^3+71x^2+100x+85$
• $y^2=11x^6+40x^5+122x^4+20x^3+80x^2+70x+53$
• $y^2=126x^6+127x^5+92x^4+51x^3+63x^2+58x+68$
• $y^2=13x^6+99x^5+57x^4+82x^3+44x^2+106x+93$
• $y^2=117x^6+95x^5+63x^4+86x^3+129x^2+106x+49$
• $y^2=86x^6+47x^5+115x^4+106x^3+80x^2+31x+62$

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 4.0.6805904.1.

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
2.137.bo_zr	$2$	(not in LMFDB)