Abelian variety isogeny class 1.121.ai over $\F_{11^{2}}$

• Label: 1.121.ai
• Base field: $\F_{11^{2}}$
• 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:	$\F_{11^{2}}$
Dimension:	$1$
L-polynomial:	$1 - 8 x + 121 x^{2}$
Frobenius angles:	$\pm0.381535076314$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-105}) \)
Galois group:	$C_2$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$114$	$14820$	$1773954$	$214356480$	$25937115954$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$114$	$14820$	$1773954$	$214356480$	$25937115954$	$3138426198180$	$379749853501314$	$45949730286520320$	$5559917314465730034$	$672749994889171270500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):

• $y^2=x^3+a^{80} x+a^{81}$
• $y^2=x^3+10 x+a^{61}$
• $y^2=x^3+a^{10} x+a^{11}$
• $y^2=x^3+a^{90} x+a^{91}$
• $y^2=x^3+a^{40} x+a^{41}$
• $y^2=x^3+a^{65} x+a^{66}$
• $y^2=x^3+a^{95} x+3$
• $y^2=x^3+a^{20} x+a^{21}$

where $a$ is a root of the Conway polynomial.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{2}}$.

Endomorphism algebra over $\F_{11^{2}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-105}) \).

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
1.121.i	$2$	(not in LMFDB)