Abelian variety isogeny class 2.163.abs_bfa over $\F_{163}$

• Label: 2.163.abs_bfa
• Base field: $\F_{163}$
• 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:	$\F_{163}$
Dimension:	$2$
L-polynomial:	$( 1 - 24 x + 163 x^{2} )( 1 - 20 x + 163 x^{2} )$
	$1 - 44 x + 806 x^{2} - 7172 x^{3} + 26569 x^{4}$
Frobenius angles:	$\pm0.110906256499$, $\pm0.213555132351$
Angle rank:	$2$ (numerical)
Jacobians:	$90$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20160$	$697374720$	$18754040652480$	$498338451704217600$	$13239725573807133508800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$120$	$26246$	$4330440$	$705950062$	$115064395800$	$18755379372854$	$3057125327382120$	$498311414782454878$	$81224760534391959480$	$13239635967025336611686$

Jacobians and polarizations

This isogeny class contains the Jacobians of 90 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=78x^6+32x^5+20x^4+27x^3+11x^2+104x+7$
• $y^2=128x^6+141x^5+49x^4+30x^3+49x^2+141x+128$
• $y^2=159x^6+36x^5+8x^4+47x^3+48x^2+156x+51$
• $y^2=21x^6+134x^5+120x^4+95x^3+120x^2+134x+21$
• $y^2=101x^6+12x^5+75x^4+47x^3+122x^2+82x+107$
• $y^2=36x^6+26x^5+10x^4+50x^3+28x^2+87x+20$
• $y^2=22x^6+62x^5+75x^4+32x^3+127x^2+104x+115$
• $y^2=12x^6+27x^5+132x^4+116x^3+114x^2+7x+105$
• $y^2=72x^6+89x^5+21x^4+55x^3+155x^2+x+5$
• $y^2=106x^6+99x^5+158x^4+84x^3+97x^2+50x+109$
• $y^2=22x^6+34x^5+93x^4+138x^3+93x^2+34x+22$
• $y^2=27x^6+142x^5+79x^4+14x^3+138x^2+19x+105$
• $y^2=149x^6+131x^5+20x^4+55x^3+107x^2+36x+70$
• $y^2=117x^6+92x^5+13x^4+113x^3+141x^2+32x+2$
• $y^2=80x^6+121x^5+67x^4+111x^3+43x^2+11x+73$
• $y^2=45x^6+159x^5+22x^4+89x^3+22x^2+159x+45$
• $y^2=82x^6+122x^5+144x^4+53x^3+118x^2+126x+152$
• $y^2=68x^6+30x^5+130x^4+121x^3+13x^2+147x+92$
• $y^2=24x^6+6x^5+39x^4+30x^3+39x^2+6x+24$
• $y^2=142x^6+128x^5+7x^4+158x^3+23x^2+18x+110$
• and

• $y^2=142x^6+60x^5+16x^4+125x^3+150x^2+88x+78$
• $y^2=117x^6+51x^5+77x^4+22x^3+33x^2+26x+42$
• $y^2=123x^6+136x^5+40x^4+78x^3+3x^2+112x+158$
• $y^2=128x^6+12x^5+110x^4+109x^3+44x^2+28x+76$
• $y^2=103x^6+117x^5+6x^4+100x^3+16x^2+17x+130$
• $y^2=136x^6+120x^5+119x^4+103x^3+142x^2+33x+147$
• $y^2=156x^6+156x^5+93x^4+149x^3+119x^2+14x+74$
• $y^2=105x^6+125x^5+98x^4+95x^3+5x^2+48x+30$
• $y^2=5x^6+49x^5+110x^4+14x^3+103x^2+112x+148$
• $y^2=117x^6+71x^5+149x^4+37x^3+142x^2+11x+60$
• $y^2=36x^6+72x^5+49x^4+151x^3+49x^2+72x+36$
• $y^2=80x^6+103x^5+141x^4+87x^3+141x^2+103x+80$
• $y^2=17x^6+82x^5+137x^4+134x^3+137x^2+82x+17$
• $y^2=7x^6+131x^5+76x^4+159x^3+76x^2+131x+7$
• $y^2=101x^6+162x^5+121x^4+10x^3+121x^2+162x+101$
• $y^2=153x^6+28x^5+33x^4+30x^3+33x^2+28x+153$
• $y^2=162x^6+7x^5+152x^4+124x^3+47x^2+4x+103$
• $y^2=122x^6+152x^5+132x^4+12x^3+132x^2+152x+122$
• $y^2=46x^6+32x^5+45x^4+64x^3+45x^2+32x+46$
• $y^2=80x^6+29x^5+78x^4+12x^3+65x^2+135x+92$
• $y^2=17x^6+85x^5+62x^4+146x^3+21x^2+51x+8$
• $y^2=3x^6+133x^5+101x^4+38x^3+89x^2+38x+149$
• $y^2=149x^6+32x^5+57x^4+127x^3+57x^2+32x+149$
• $y^2=107x^6+41x^5+81x^4+18x^3+57x^2+152x+116$
• $y^2=4x^6+57x^5+23x^4+17x^3+23x^2+57x+4$
• $y^2=3x^6+47x^5+139x^4+155x^3+139x^2+47x+3$
• $y^2=2x^6+95x^5+90x^4+157x^3+5x^2+80x+94$
• $y^2=39x^6+138x^5+16x^4+20x^3+58x^2+107x+95$
• $y^2=46x^6+106x^5+100x^4+106x^3+144x^2+128x+87$
• $y^2=5x^6+154x^5+102x^4+45x^3+102x^2+154x+5$
• $y^2=96x^6+121x^5+143x^4+x^3+143x^2+121x+96$
• $y^2=72x^6+142x^5+23x^4+2x^3+68x^2+131x+77$
• $y^2=137x^6+117x^5+9x^4+90x^3+9x^2+117x+137$
• $y^2=159x^6+40x^5+13x^4+114x^3+84x^2+7x+23$
• $y^2=2x^6+78x^5+89x^4+46x^3+47x^2+11x+123$
• $y^2=36x^6+46x^5+30x^4+137x^3+30x^2+46x+36$
• $y^2=72x^6+79x^5+138x^4+49x^3+21x^2+50x+86$
• $y^2=25x^6+127x^5+119x^4+145x^3+119x^2+127x+25$
• $y^2=139x^6+18x^5+52x^4+74x^3+14x^2+131x+88$
• $y^2=75x^6+29x^5+131x^4+121x^3+73x^2+125x+147$
• $y^2=102x^6+134x^5+52x^4+54x^3+81x^2+18x+88$
• $y^2=152x^6+64x^5+14x^4+112x^3+14x^2+64x+152$
• $y^2=149x^6+36x^5+125x^4+22x^3+12x^2+51x+32$
• $y^2=160x^6+127x^4+19x^3+137x^2+88$
• $y^2=68x^6+5x^5+53x^4+119x^3+59x^2+101$
• $y^2=157x^6+89x^5+83x^4+12x^3+96x^2+76x+86$
• $y^2=50x^6+5x^5+29x^4+92x^3+155x^2+129x+21$
• $y^2=38x^6+57x^5+25x^4+143x^3+25x^2+57x+38$
• $y^2=123x^6+119x^5+101x^4+56x^3+101x^2+119x+123$
• $y^2=141x^6+110x^5+4x^4+49x^3+4x^2+110x+141$
• $y^2=72x^6+103x^5+66x^4+8x^3+82x^2+109x+122$
• $y^2=129x^6+33x^5+105x^4+124x^3+3x^2+29x+10$
• $y^2=73x^6+143x^5+8x^4+52x^3+8x^2+143x+73$
• $y^2=37x^6+60x^5+156x^4+3x^3+88x^2+85x+125$
• $y^2=79x^6+54x^5+122x^4+99x^3+159x^2+160x+107$
• $y^2=157x^6+83x^5+47x^4+87x^3+134x^2+158x+5$
• $y^2=139x^6+71x^5+147x^4+31x^3+107x^2+100x+17$
• $y^2=159x^6+130x^5+13x^4+44x^3+13x^2+130x+159$
• $y^2=124x^6+125x^5+4x^4+37x^3+156x^2+67x+18$
• $y^2=137x^6+28x^5+69x^4+144x^3+85x^2+82x+80$
• $y^2=67x^6+91x^5+21x^4+82x^3+133x^2+46x+153$
• $y^2=162x^6+108x^5+88x^4+102x^3+131x^2+52x+105$
• $y^2=105x^6+4x^5+81x^4+16x^3+81x^2+4x+105$
• $y^2=148x^6+48x^5+76x^4+98x^3+73x^2+113x+99$
• $y^2=113x^6+144x^5+131x^4+49x^3+130x^2+135x+32$
• $y^2=70x^6+28x^5+33x^4+142x^3+36x^2+127x+68$
• $y^2=59x^6+129x^5+106x^4+158x^3+106x^2+129x+59$
• $y^2=31x^6+161x^5+62x^4+53x^3+15x^2+21x+125$
• $y^2=79x^6+57x^5+75x^4+118x^3+75x^2+57x+79$
• $y^2=67x^5+141x^4+41x^3+19x^2+20x$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{163}$.

Endomorphism algebra over $\F_{163}$

The isogeny class factors as 1.163.ay $\times$ 1.163.au and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.163.ay : \(\Q(\sqrt{-19}) \).
• 1.163.au : \(\Q(\sqrt{-7}) \).

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
2.163.ae_afy	$2$	(not in LMFDB)
2.163.e_afy	$2$	(not in LMFDB)
2.163.bs_bfa	$2$	(not in LMFDB)