Abelian variety isogeny class 1.81.an over $\F_{3^{4}}$

• Label: 1.81.an
• Base field: $\F_{3^{4}}$
• 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:	$\F_{3^{4}}$
Dimension:	$1$
L-polynomial:	$1 - 13 x + 81 x^{2}$
Frobenius angles:	$\pm0.243120792737$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-155}) \)
Galois group:	$C_2$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$69$	$6555$	$532404$	$43059795$	$3486876429$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$69$	$6555$	$532404$	$43059795$	$3486876429$	$282429673920$	$22876786787469$	$1853020104041955$	$150094634653537524$	$12157665457561528875$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 0 are hyperelliptic):

• $y^2+x y=x^3+a^2$
• $y^2+x y=x^3+a^{54}$
• $y^2+x y=x^3+a^{18}$
• $y^2+x y=x^3+a^6$

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{4}}$.

Endomorphism algebra over $\F_{3^{4}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-155}) \).

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
1.81.n	$2$	(not in LMFDB)