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

• Label: 1.81.ar
• 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:	$1 - 17 x + 81 x^{2}$
Frobenius angles:	$\pm0.106600758076$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-35}) \)
Galois group:	$C_2$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$65$	$6435$	$530660$	$43043715$	$3486796625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$65$	$6435$	$530660$	$43043715$	$3486796625$	$282429987840$	$22876799137985$	$1853020265903235$	$150094636065547940$	$12157665465881095875$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):

• $y^2+x y=x^3+a^{38} x^2+a^{49}$
• $y^2+x y=x^3+a^{77} x^2+a^{23}$

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 \(\Q(\sqrt{-35}) \).

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.r	$2$	(not in LMFDB)