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

• Label: 2.137.abo_zp
• 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:	$( 1 - 23 x + 137 x^{2} )( 1 - 17 x + 137 x^{2} )$
	$1 - 40 x + 665 x^{2} - 5480 x^{3} + 18769 x^{4}$
Frobenius angles:	$\pm0.0596181899068$, $\pm0.241282780251$
Angle rank:	$2$ (numerical)
Jacobians:	$44$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13915$	$347248825$	$6610210097920$	$124101391104315625$	$2329198381471590022075$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$18500$	$2570714$	$352285188$	$48261814258$	$6611854868750$	$905824257200674$	$124097929228986628$	$17001416399439692618$	$2329194047566011402500$

Jacobians and polarizations

This isogeny class contains the Jacobians of 44 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=14x^6+67x^5+66x^4+129x^3+66x^2+67x+14$
• $y^2=4x^6+64x^5+99x^4+74x^3+99x^2+64x+4$
• $y^2=120x^6+82x^5+18x^4+130x^3+94x^2+74x+42$
• $y^2=13x^6+10x^5+107x^4+58x^3+99x^2+9x+72$
• $y^2=93x^6+27x^5+6x^4+25x^3+6x^2+27x+93$
• $y^2=111x^6+34x^5+44x^4+81x^3+36x^2+79x+47$
• $y^2=41x^6+34x^5+36x^4+134x^3+50x^2+2x+88$
• $y^2=103x^6+118x^5+28x^4+19x^3+28x^2+118x+103$
• $y^2=128x^6+121x^5+37x^4+53x^3+60x^2+74x+98$
• $y^2=47x^6+12x^5+58x^4+119x^3+123x^2+39x+122$
• $y^2=42x^6+4x^5+55x^4+23x^3+17x^2+121x+37$
• $y^2=41x^6+41x^4+37x^3+69x^2+78x+83$
• $y^2=89x^6+129x^5+75x^4+102x^3+5x^2+10x+126$
• $y^2=125x^6+125x^5+86x^4+67x^3+68x^2+18x+94$
• $y^2=130x^6+24x^5+33x^4+42x^3+33x^2+24x+130$
• $y^2=75x^6+67x^5+91x^4+18x^3+91x^2+67x+75$
• $y^2=71x^6+97x^5+85x^4+130x^3+85x^2+97x+71$
• $y^2=51x^6+111x^5+136x^4+70x^3+35x^2+23x+79$
• $y^2=9x^6+59x^5+99x^4+127x^3+41x^2+19x+23$
• $y^2=61x^6+24x^5+71x^4+24x^3+120x^2+65x+89$
• and

• $y^2=29x^6+95x^5+22x^4+48x^3+22x^2+95x+29$
• $y^2=93x^6+4x^5+103x^4+19x^3+38x^2+136x+42$
• $y^2=20x^6+112x^5+122x^4+92x^3+122x^2+112x+20$
• $y^2=94x^6+80x^5+113x^4+81x^3+21x^2+74x+65$
• $y^2=73x^6+122x^5+62x^4+96x^3+46x^2+17x+112$
• $y^2=99x^6+43x^5+83x^4+106x^3+16x^2+13x+95$
• $y^2=111x^6+33x^4+9x^2+19x+51$
• $y^2=49x^6+2x^5+128x^4+82x^3+101x^2+16x+45$
• $y^2=108x^6+78x^5+86x^4+133x^3+55x^2+100x+59$
• $y^2=30x^6+19x^5+101x^4+28x^3+51x^2+89x+57$
• $y^2=28x^6+41x^5+135x^4+98x^3+41x^2+46x+24$
• $y^2=57x^6+3x^5+120x^4+133x^3+120x^2+3x+57$
• $y^2=116x^6+128x^5+50x^4+13x^3+50x^2+128x+116$
• $y^2=95x^6+55x^5+116x^4+124x^3+126x^2+79x+35$
• $y^2=19x^6+91x^5+113x^4+73x^3+113x^2+91x+19$
• $y^2=117x^6+108x^5+133x^4+68x^3+5x^2+96x+24$
• $y^2=135x^6+129x^5+79x^4+117x^3+78x^2+37x+76$
• $y^2=11x^6+99x^5+103x^4+21x^3+78x^2+48x+122$
• $y^2=29x^6+40x^5+97x^4+29x^3+8x^2+105x+60$
• $y^2=118x^6+80x^5+82x^4+57x^3+34x^2+119x+55$
• $y^2=54x^6+15x^5+123x^4+109x^3+121x^2+39x+58$
• $y^2=21x^6+10x^5+119x^4+33x^3+54x^2+124x+13$
• $y^2=14x^6+68x^5+132x^4+48x^3+103x^2+87x+56$
• $y^2=100x^6+109x^5+86x^4+40x^3+112x^2+97x+37$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{137}$

The isogeny class factors as 1.137.ax $\times$ 1.137.ar and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.137.ax : \(\Q(\sqrt{-19}) \).
• 1.137.ar : \(\Q(\sqrt{-259}) \).

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.ag_aen	$2$	(not in LMFDB)
2.137.g_aen	$2$	(not in LMFDB)
2.137.bo_zp	$2$	(not in LMFDB)