Abelian variety isogeny class 2.193.abw_bkr over $\F_{193}$

• Label: 2.193.abw_bkr
• Base field: $\F_{193}$
• 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:	$\F_{193}$
Dimension:	$2$
L-polynomial:	$( 1 - 27 x + 193 x^{2} )( 1 - 21 x + 193 x^{2} )$
	$1 - 48 x + 953 x^{2} - 9264 x^{3} + 37249 x^{4}$
Frobenius angles:	$\pm0.0758389534121$, $\pm0.227245037274$
Angle rank:	$2$ (numerical)
Jacobians:	$33$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$28891$	$1372755865$	$51674261395648$	$1925162249045013225$	$71709054349001000898571$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$146$	$36852$	$7187906$	$1387516324$	$267785742386$	$51682544504574$	$9974730289189490$	$1925122951498529476$	$371548729896116198978$	$71708904873316415192532$

Jacobians and polarizations

This isogeny class contains the Jacobians of 33 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=185x^6+136x^5+154x^4+191x^3+27x^2+146x+37$
• $y^2=41x^6+123x^5+160x^4+29x^3+13x^2+2x+189$
• $y^2=88x^6+153x^5+35x^4+131x^3+103x^2+125x+142$
• $y^2=53x^6+118x^5+94x^4+25x^3+94x^2+118x+53$
• $y^2=73x^6+36x^5+166x^4+126x^3+141x^2+146x+5$
• $y^2=111x^6+x^5+149x^4+138x^3+150x^2+25x+165$
• $y^2=127x^6+150x^5+26x^4+144x^3+26x^2+150x+127$
• $y^2=91x^6+92x^5+48x^4+145x^3+147x^2+137x+50$
• $y^2=9x^6+179x^5+173x^4+41x^3+173x^2+179x+9$
• $y^2=72x^6+112x^5+82x^4+97x^3+82x^2+112x+72$
• $y^2=49x^6+183x^5+162x^4+131x^3+104x^2+70x+185$
• $y^2=10x^6+190x^5+32x^4+168x^3+189x^2+175x+163$
• $y^2=75x^6+168x^5+88x^4+88x^3+8x^2+58x+12$
• $y^2=172x^6+137x^5+159x^4+37x^3+159x^2+137x+172$
• $y^2=2x^6+94x^5+116x^4+166x^3+79x^2+15x+60$
• $y^2=112x^6+143x^5+17x^4+99x^3+56x^2+109x+98$
• $y^2=142x^6+98x^5+39x^4+90x^3+92x^2+7x+104$
• $y^2=4x^6+138x^5+15x^4+9x^3+129x^2+73x+88$
• $y^2=121x^6+86x^5+122x^4+6x^3+16x^2+90x+160$
• $y^2=121x^6+58x^5+90x^4+48x^3+77x^2+118x+1$
• and

• $y^2=53x^6+106x^5+24x^4+143x^3+151x^2+40x+140$
• $y^2=60x^6+23x^5+184x^4+24x^3+118x^2+72x+85$
• $y^2=133x^6+147x^5+73x^4+138x^3+73x^2+147x+133$
• $y^2=171x^6+64x^5+58x^4+95x^3+85x^2+189x+8$
• $y^2=66x^6+120x^5+53x^4+84x^3+53x^2+120x+66$
• $y^2=55x^6+118x^5+183x^4+133x^3+183x^2+118x+55$
• $y^2=81x^6+102x^5+27x^4+103x^3+127x^2+148x+102$
• $y^2=131x^6+85x^5+29x^4+11x^3+29x^2+85x+131$
• $y^2=87x^6+114x^5+39x^4+96x^3+80x^2+99x+61$
• $y^2=25x^6+180x^5+170x^4+94x^3+3x^2+160x+162$
• $y^2=29x^6+29x^5+104x^4+187x^3+61x^2+21$
• $y^2=17x^6+171x^5+125x^4+32x^3+10x^2+26x+160$
• $y^2=63x^6+50x^5+155x^4+99x^3+73x^2+94x+135$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{193}$.

Endomorphism algebra over $\F_{193}$

The isogeny class factors as 1.193.abb $\times$ 1.193.av and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.193.abb : \(\Q(\sqrt{-43}) \).
• 1.193.av : \(\Q(\sqrt{-331}) \).

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
2.193.ag_agz	$2$	(not in LMFDB)
2.193.g_agz	$2$	(not in LMFDB)
2.193.bw_bkr	$2$	(not in LMFDB)