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

• Label: 2.137.abo_zo
• 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:	$1 - 40 x + 664 x^{2} - 5480 x^{3} + 18769 x^{4}$
Frobenius angles:	$\pm0.0462922860036$, $\pm0.244475578969$
Angle rank:	$2$ (numerical)
Number field:	4.0.4870400.2
Galois group:	$D_{4}$
Jacobians:	$16$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13914$	$347209956$	$6609900924666$	$124100072741043216$	$2329194443295623169114$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$18498$	$2570594$	$352281446$	$48261732658$	$6611853479202$	$905824237637074$	$124097928990786238$	$17001416396769602018$	$2329194047535506042178$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=58x^6+77x^5+51x^4+109x^3+127x^2+112x+104$
• $y^2=27x^6+135x^5+76x^4+16x^3+74x^2+25x+46$
• $y^2=48x^6+33x^5+85x^4+75x^2+111x+77$
• $y^2=66x^6+101x^5+52x^4+120x^3+130x^2+16x+21$
• $y^2=41x^6+8x^5+48x^4+45x^3+44x^2+109x+110$
• $y^2=33x^6+13x^5+27x^4+13x^3+56x^2+45x+21$
• $y^2=18x^6+58x^5+104x^4+57x^3+56x^2+40x+134$
• $y^2=51x^6+93x^5+87x^4+x^3+96x^2+58x+117$
• $y^2=67x^6+50x^5+62x^4+39x^3+35x^2+134x+69$
• $y^2=50x^6+58x^5+98x^4+41x^3+58x^2+54x+111$
• $y^2=82x^6+115x^5+12x^4+134x^3+3x^2+20x+13$
• $y^2=58x^6+12x^5+100x^4+130x^3+91x^2+4x+11$
• $y^2=48x^6+45x^5+114x^4+13x^3+114x^2+79x+80$
• $y^2=16x^6+82x^5+61x^4+22x^3+69x^2+109x+83$
• $y^2=41x^6+124x^5+49x^4+89x^3+3x^2+28x+29$
• $y^2=79x^6+121x^5+109x^4+85x^3+102x^2+110x+121$

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 4.0.4870400.2.

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.bo_zo	$2$	(not in LMFDB)