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

• Label: 2.137.abo_zy
• Base field: $\F_{137}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{137}$
Dimension:	$2$
L-polynomial:	$( 1 - 20 x + 137 x^{2} )^{2}$
	$1 - 40 x + 674 x^{2} - 5480 x^{3} + 18769 x^{4}$
Frobenius angles:	$\pm0.173950867407$, $\pm0.173950867407$
Angle rank:	$1$ (numerical)
Jacobians:	$13$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13924$	$347598736$	$6612992837476$	$124113193119256576$	$2329232956424663043364$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$18518$	$2571794$	$352318686$	$48262530658$	$6611866439222$	$905824399656274$	$124097930438297278$	$17001416402199314018$	$2329194047431459622678$

Jacobians and polarizations

This isogeny class contains the Jacobians of 13 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=70x^6+73x^5+34x^4+60x^3+7x^2+25x+85$
• $y^2=108x^6+29x^5+108x^4+123x^3+108x^2+29x+108$
• $y^2=44x^6+31x^5+99x^4+126x^3+27x^2+74x+23$
• $y^2=25x^6+8x^5+47x^4+63x^3+116x^2+48x+33$
• $y^2=63x^6+98x^5+89x^4+121x^3+55x^2+71x+129$
• $y^2=11x^6+15x^5+101x^4+117x^3+122x^2+109x+92$
• $y^2=51x^6+82x^5+80x^4+111x^3+23x^2+56x+11$
• $y^2=96x^6+132x^5+135x^4+94x^3+135x^2+132x+96$
• $y^2=86x^6+67x^5+81x^4+72x^3+72x^2+47x+27$
• $y^2=89x^6+43x^5+49x^4+36x^3+81x^2+117x+111$
• $y^2=76x^6+51x^5+124x^4+11x^3+116x^2+16x+1$
• $y^2=91x^6+117x^4+117x^2+91$
• $y^2=104x^6+118x^5+61x^4+105x^3+77x^2+130x+7$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{137}$

The isogeny class factors as 1.137.au^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-37}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.137.a_aew	$2$	(not in LMFDB)
2.137.bo_zy	$2$	(not in LMFDB)
2.137.u_kd	$3$	(not in LMFDB)