Abelian variety isogeny class 2.27.au_fy over $\F_{3^{3}}$

• Label: 2.27.au_fy
• Base field: $\F_{3^{3}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{3}}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 27 x^{2} )^{2}$
	$1 - 20 x + 154 x^{2} - 540 x^{3} + 729 x^{4}$
Frobenius angles:	$\pm0.0877398280459$, $\pm0.0877398280459$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$324$	$467856$	$380016036$	$281731654656$	$205849551061764$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$638$	$19304$	$530126$	$14346008$	$387427022$	$10460496824$	$282430796318$	$7625606205608$	$205891185285278$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=ax^6+(2a^2+2)x^5+ax^4+ax^2+(a^2+1)x+a$

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.27.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{3}}$.

Subfield	Primitive Model
$\F_{3}$	2.3.ac_b
$\F_{3}$	2.3.e_k

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.27.a_abu	$2$	2.729.ado_fhm
2.27.u_fy	$2$	2.729.ado_fhm
2.27.k_cv	$3$	(not in LMFDB)