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

• Label: 2.27.an_dm
• 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: yes

Invariants

Base field:	$\F_{3^{3}}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x + 27 x^{2} )( 1 - 4 x + 27 x^{2} )$
	$1 - 13 x + 90 x^{2} - 351 x^{3} + 729 x^{4}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.374235869875$
Angle rank:	$1$ (numerical)
Jacobians:	$6$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$456$	$539904$	$392577696$	$282825470976$	$205885332491736$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$741$	$19944$	$532185$	$14348505$	$387431622$	$10460605443$	$282431130609$	$7625599708248$	$205891097946261$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=(2a^2+1)x^6+(a^2+a)x^4+2ax^3+2a^2x+2a^2+2a+1$
• $y^2=(a^2+a+2)x^6+x^4+ax^3+(2a^2+a+2)x+2a^2+a$
• $y^2=(a^2+2)x^6+(a^2+a+1)x^4+(2a^2+2a)x^3+(a^2+2a+1)x+2a^2+a$
• $y^2=(2a^2+a+2)x^6+x^4+(a^2+1)x^3+(a^2+a)x+2a^2+1$
• $y^2=(a^2+a+1)x^6+(2a^2+a+1)x^4+(2a+1)x^3+(a^2+a+2)x$
• $y^2=(2a^2+a+1)x^6+(2a+2)x^4+(a^2+2a+2)x^3+(2a^2+a)x$

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.ae 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.ae : \(\Q(\sqrt{-23}) \).

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

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

• 1.387420489.abpty : \(\Q(\sqrt{-23}) \).
• 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.bm. The endomorphism algebra for each factor is:

  • 1.729.abb : \(\Q(\sqrt{-3}) \).
  • 1.729.bm : \(\Q(\sqrt{-23}) \).
• Endomorphism algebra over $\F_{3^{9}}$
  The base change of $A$ to $\F_{3^{9}}$ is 1.19683.a $\times$ 1.19683.ka. The endomorphism algebra for each factor is:

  • 1.19683.a : \(\Q(\sqrt{-3}) \).
  • 1.19683.ka : \(\Q(\sqrt{-23}) \).

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.af_s	$2$	2.729.l_qq
2.27.f_s	$2$	2.729.l_qq
2.27.n_dm	$2$	2.729.l_qq
2.27.ae_cc	$3$	(not in LMFDB)
2.27.f_s	$3$	(not in LMFDB)