Abelian variety isogeny class 1.223.at over $\F_{223}$

• Label: 1.223.at
• Base field: $\F_{223}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{223}$
Dimension:	$1$
L-polynomial:	$1 - 19 x + 223 x^{2}$
Frobenius angles:	$\pm0.280518908343$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-59}) \)
Galois group:	$C_2$
Jacobians:	$9$
Isomorphism classes:	9

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$1$
Slopes:	$[0, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$205$	$49815$	$11095420$	$2473065675$	$551473524775$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$205$	$49815$	$11095420$	$2473065675$	$551473524775$	$122978484180720$	$27424204334144305$	$6115597636330401075$	$1363778273701496439220$	$304122555035061211594575$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which 0 are hyperelliptic):

• $y^2=x^3+47 x+47$
• $y^2=x^3+221 x+217$
• $y^2=x^3+139 x+139$
• $y^2=x^3+210 x+184$
• $y^2=x^3+70 x+70$
• $y^2=x^3+56 x+168$
• $y^2=x^3+30 x+90$
• $y^2=x^3+204 x+204$
• $y^2=x^3+82 x+23$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{223}$.

Endomorphism algebra over $\F_{223}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-59}) \).

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
1.223.t	$2$	(not in LMFDB)