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

• Label: 2.137.abn_zd
• 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:	$1 - 39 x + 653 x^{2} - 5343 x^{3} + 18769 x^{4}$
Frobenius angles:	$\pm0.157033176591$, $\pm0.212540234294$
Angle rank:	$2$ (numerical)
Number field:	4.0.645525.5
Galois group:	$D_{4}$
Jacobians:	$24$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14041$	$348287005$	$6614568638329$	$124114897097925525$	$2329231398328468877056$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$99$	$18555$	$2572407$	$352323523$	$48262498374$	$6611864645715$	$905824365935847$	$124097930041270243$	$17001416399851251819$	$2329194047453108559150$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=13x^6+44x^5+32x^4+45x^3+85x^2+6x+70$
• $y^2=14x^6+66x^5+38x^4+114x^3+28x^2+122x+49$
• $y^2=41x^6+61x^5+71x^4+41x^3+124x^2+96x+107$
• $y^2=88x^6+89x^5+98x^4+50x^3+55x^2+107x+107$
• $y^2=2x^6+94x^5+65x^4+54x^3+132x+97$
• $y^2=33x^6+65x^5+72x^4+120x^3+70x^2+49x+57$
• $y^2=18x^6+106x^5+97x^4+93x^3+x^2+122x+68$
• $y^2=125x^6+53x^5+38x^4+39x^3+18x^2+88x+73$
• $y^2=8x^6+14x^5+73x^4+57x^3+29x^2+8x+39$
• $y^2=55x^6+12x^5+102x^4+135x^3+117x^2+94x+99$
• $y^2=50x^6+102x^4+62x^3+87x^2+135x+106$
• $y^2=32x^6+29x^5+29x^4+63x^3+111x^2+49x+86$
• $y^2=74x^6+6x^5+50x^4+11x^3+121x^2+71x+75$
• $y^2=10x^6+85x^5+134x^4+32x^3+105x^2+69x+7$
• $y^2=26x^6+105x^5+18x^4+110x^3+85x^2+131x+82$
• $y^2=83x^6+74x^5+95x^4+62x^3+88x^2+38x+61$
• $y^2=91x^6+118x^5+125x^4+114x^3+81x^2+106x+127$
• $y^2=90x^6+34x^5+39x^4+49x^3+15x^2+53x+90$
• $y^2=9x^6+57x^5+18x^4+110x^3+109x^2+35x+133$
• $y^2=91x^6+87x^5+16x^4+120x^3+25x^2+29x+43$
• $y^2=83x^6+14x^5+95x^4+16x^3+117x^2+71x+22$
• $y^2=132x^6+10x^5+82x^4+20x^3+54x^2+98x+82$
• $y^2=28x^6+2x^5+30x^4+18x^3+103x^2+130x+75$
• $y^2=12x^6+118x^5+126x^4+134x^3+14x^2+108x+68$

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 4.0.645525.5.

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_zd	$2$	(not in LMFDB)