Abelian variety isogeny class 2.199.aby_bna over $\F_{199}$

• Label: 2.199.aby_bna
• Base field: $\F_{199}$
• 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:	$\F_{199}$
Dimension:	$2$
L-polynomial:	$( 1 - 28 x + 199 x^{2} )( 1 - 22 x + 199 x^{2} )$
	$1 - 50 x + 1014 x^{2} - 9950 x^{3} + 39601 x^{4}$
Frobenius angles:	$\pm0.0391815390403$, $\pm0.215336333813$
Angle rank:	$2$ (numerical)
Jacobians:	$28$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$30616$	$1549659456$	$62082171693304$	$2459377344975243264$	$97393731271080446549176$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$150$	$39130$	$7877850$	$1568241214$	$312079773750$	$62103838525306$	$12358664129240250$	$2459374187798771134$	$489415464055900886550$	$97393677358933863237850$

Jacobians and polarizations

This isogeny class contains the Jacobians of 28 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=101x^6+6x^5+122x^4+94x^3+85x^2+41x+89$
• $y^2=6x^6+16x^5+90x^4+22x^3+32x^2+9x+41$
• $y^2=92x^6+63x^5+142x^4+173x^3+191x^2+160x+61$
• $y^2=8x^6+104x^5+97x^4+136x^3+90x^2+93x+27$
• $y^2=118x^6+35x^5+42x^4+154x^3+28x^2+105x+47$
• $y^2=45x^6+56x^5+126x^4+41x^3+195x^2+165x+196$
• $y^2=34x^6+192x^5+43x^4+152x^3+181x^2+47x+178$
• $y^2=29x^6+146x^5+155x^4+181x^3+34x^2+7x+89$
• $y^2=32x^6+15x^5+169x^4+41x^3+132x^2+50x+91$
• $y^2=23x^6+33x^5+98x^4+115x^3+127x^2+116x+114$
• $y^2=157x^6+18x^5+61x^4+33x^3+153x^2+163x+3$
• $y^2=193x^6+15x^5+136x^4+162x^3+89x^2+57x+30$
• $y^2=127x^6+144x^5+125x^4+112x^3+28x^2+185x+150$
• $y^2=74x^6+39x^5+69x^4+45x^3+91x^2+47x+129$
• $y^2=80x^6+146x^5+42x^4+137x^3+42x^2+146x+80$
• $y^2=67x^6+176x^5+28x^4+26x^3+96x^2+36x+38$
• $y^2=191x^6+119x^5+105x^4+25x^3+105x^2+119x+191$
• $y^2=189x^6+28x^5+81x^4+19x^3+81x^2+28x+189$
• $y^2=24x^6+74x^5+22x^4+184x^3+152x^2+82x+84$
• $y^2=66x^6+175x^5+113x^4+116x^3+115x^2+143x+125$
• $y^2=85x^6+156x^5+120x^4+6x^3+100x^2+50x+74$
• $y^2=33x^6+123x^5+165x^4+192x^3+13x^2+130x+107$
• $y^2=33x^6+111x^5+163x^4+11x^3+163x^2+111x+33$
• $y^2=169x^6+118x^5+5x^4+30x^3+6x^2+98x+99$
• $y^2=13x^6+48x^5+153x^4+155x^3+92x^2+142x+51$
• $y^2=154x^6+87x^5+65x^4+4x^3+142x^2+175x+119$
• $y^2=96x^6+115x^5+115x^4+107x^3+128x^2+184x+181$
• $y^2=37x^6+71x^5+68x^4+175x^3+133x^2+151x+157$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{199}$.

Endomorphism algebra over $\F_{199}$

The isogeny class factors as 1.199.abc $\times$ 1.199.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.199.abc : \(\Q(\sqrt{-3}) \).
• 1.199.aw : \(\Q(\sqrt{-78}) \).

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
2.199.ag_aik	$2$	(not in LMFDB)
2.199.g_aik	$2$	(not in LMFDB)
2.199.by_bna	$2$	(not in LMFDB)
2.199.al_ga	$3$	(not in LMFDB)
2.199.af_y	$3$	(not in LMFDB)