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

• Label: 2.193.aby_bmx
• 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:	$( 1 - 25 x + 193 x^{2} )^{2}$
	$1 - 50 x + 1011 x^{2} - 9650 x^{3} + 37249 x^{4}$
Frobenius angles:	$\pm0.143734387197$, $\pm0.143734387197$
Angle rank:	$1$ (numerical)
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$28561$	$1369814121$	$51666021416464$	$1925171176753933641$	$71709256409245886243761$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$144$	$36772$	$7186758$	$1387522756$	$267786496944$	$51682566660478$	$9974730725425008$	$1925122957865007748$	$371548729959925111494$	$71708904873488421516772$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=5 x^6+133 x^3+45$
• $y^2=20 x^6+119 x^5+92 x^4+19 x^3+190 x^2+10 x+106$
• $y^2=75 x^6+177 x^5+192 x^4+64 x^3+95 x^2+92 x+111$
• $y^2=127 x^6+42 x^5+95 x^4+168 x^3+x^2+26 x+73$
• $y^2=126 x^6+82 x^5+106 x^4+7 x^3+45 x^2+186 x+94$
• $y^2=10 x^6+56 x^5+39 x^4+30 x^3+40 x^2+107 x+91$
• $y^2=119 x^6+106 x^5+50 x^4+115 x^3+112 x^2+119 x+160$
• $y^2=53 x^6+144 x^5+138 x^4+156 x^3+108 x^2+x+37$
• $y^2=28 x^6+92 x^5+60 x^4+170 x^3+81 x^2+56 x+102$
• $y^2=89 x^6+91 x^5+93 x^4+125 x^3+24 x^2+88 x+87$
• $y^2=12 x^6+98 x^5+115 x^4+48 x^3+151 x^2+77 x+98$
• $y^2=155 x^6+117 x^5+83 x^4+69 x^3+166 x^2+82 x+82$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{193}$

The isogeny class factors as 1.193.az^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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
2.193.a_ajf	$2$	(not in LMFDB)
2.193.by_bmx	$2$	(not in LMFDB)
2.193.ax_my	$3$	(not in LMFDB)
2.193.ac_ahh	$3$	(not in LMFDB)
2.193.e_pa	$3$	(not in LMFDB)
2.193.z_qq	$3$	(not in LMFDB)
2.193.bu_bjf	$3$	(not in LMFDB)