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

• Label: 1.121.ao
• Base field: $\F_{11^{2}}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{11^{2}}$
Dimension:	$1$
L-polynomial:	$1 - 14 x + 121 x^{2}$
Frobenius angles:	$\pm0.280437798008$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}) \)
Galois group:	$C_2$
Jacobians:	$9$
Isomorphism classes:	9

This isogeny class is simple and geometrically simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$108$	$14688$	$1773900$	$214386048$	$25937522028$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$108$	$14688$	$1773900$	$214386048$	$25937522028$	$3138426453600$	$379749794870988$	$45949729554298368$	$5559917313846581100$	$672749994974943032928$

Jacobians and polarizations

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

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 \(\Q(\sqrt{-2}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{11^{2}}$.

Subfield	Primitive Model
$\F_{11}$	1.11.ag
$\F_{11}$	1.11.g

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
1.121.o	$2$	(not in LMFDB)