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

• Label: 2.27.ar_ew
• Base field: $\F_{3^{3}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3^{3}}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x + 27 x^{2} )( 1 - 8 x + 27 x^{2} )$
	$1 - 17 x + 126 x^{2} - 459 x^{3} + 729 x^{4}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.220355751984$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$380$	$506160$	$390136880$	$283540708800$	$206088548510900$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$693$	$19820$	$533529$	$14362661$	$387480726$	$10460503127$	$282429286641$	$7625591969780$	$205891094718693$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{18}}$.

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

The isogeny class factors as 1.27.aj $\times$ 1.27.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.27.aj : \(\Q(\sqrt{-3}) \).
• 1.27.ai : \(\Q(\sqrt{-11}) \).

Endomorphism algebra over $\overline{\F}_{3^{3}}$

The base change of $A$ to $\F_{3^{18}}$ is 1.387420489.bews $\times$ 1.387420489.cggc. The endomorphism algebra for each factor is:

• 1.387420489.bews : \(\Q(\sqrt{-11}) \).
• 1.387420489.cggc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{6}}$
  The base change of $A$ to $\F_{3^{6}}$ is 1.729.abb $\times$ 1.729.ak. The endomorphism algebra for each factor is:

  • 1.729.abb : \(\Q(\sqrt{-3}) \).
  • 1.729.ak : \(\Q(\sqrt{-11}) \).
• Endomorphism algebra over $\F_{3^{9}}$
  The base change of $A$ to $\F_{3^{9}}$ is 1.19683.a $\times$ 1.19683.fg. The endomorphism algebra for each factor is:

  • 1.19683.a : \(\Q(\sqrt{-3}) \).
  • 1.19683.fg : \(\Q(\sqrt{-11}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.27.ab_as	$2$	2.729.abl_com
2.27.b_as	$2$	2.729.abl_com
2.27.r_ew	$2$	2.729.abl_com
2.27.ai_cc	$3$	(not in LMFDB)