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

• Label: 2.27.am_di
• Base field: $\F_{3^{3}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• 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:	$( 1 - 8 x + 27 x^{2} )( 1 - 4 x + 27 x^{2} )$
	$1 - 12 x + 86 x^{2} - 324 x^{3} + 729 x^{4}$
Frobenius angles:	$\pm0.220355751984$, $\pm0.374235869875$
Angle rank:	$2$ (numerical)
Jacobians:	$54$
Isomorphism classes:	258

This isogeny class is not simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$480$	$552960$	$395290080$	$283159756800$	$205894382258400$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$758$	$20080$	$532814$	$14349136$	$387413126$	$10460401072$	$282429817886$	$7625594193040$	$205891089268118$

Jacobians and polarizations

This isogeny class contains the Jacobians of 54 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+(2a^2+a+1)x^3+(a^2+a)x^2+(2a^2+2a)x+a^2+1$
• $y^2=(2a^2+2a)x^6+(a+1)x^4+(a^2+a)x^3+x^2+(2a^2+2a)x+a^2+a+2$
• $y^2=(a^2+a+2)x^6+(2a+1)x^5+(2a+1)x^4+(a^2+a+1)x^3+a^2x^2+2a^2x+a+1$
• $y^2=(2a^2+2a+2)x^6+(a^2+a)x^5+(a^2+1)x^4+(a^2+a)x^3+(2a^2+2a+1)x^2+(a+2)x+a+2$
• $y^2=2ax^6+2x^5+(2a^2+1)x^4+(2a^2+2a+1)x^3+(2a^2+1)x^2+2x+2a$
• $y^2=(a^2+2a+2)x^6+(2a^2+a+2)x^5+(a^2+a+1)x^4+(a^2+2a+2)x^3+a^2x^2+(2a^2+2a+2)x+2a^2$
• $y^2=(2a^2+a)x^6+(a^2+a+1)x^5+(a^2+a)x^3+(a^2+a+1)x+2a^2+a$
• $y^2=(2a+2)x^6+ax^5+(a^2+2a)x^4+(2a^2+a+2)x^3+(a^2+2a)x^2+(a^2+1)x+a^2+2a$
• $y^2=(2a^2+a)x^6+(a^2+a)x^5+x^4+2a^2x^3+(2a^2+2a+1)x^2+(a^2+a+1)x+a^2+a+2$
• $y^2=(2a^2+2a)x^6+2a^2x^5+ax^4+(2a^2+a+2)x^3+2x^2+(a^2+2)x+a^2+a$
• $y^2=(a+2)x^6+(a^2+a)x^5+(a^2+2)x^4+(a^2+1)x^3+(a^2+2)x^2+(a^2+a)x+a+2$
• $y^2=2a^2x^6+(2a+1)x^5+(a+1)x^4+2a^2x^3+(a^2+a+2)x^2+(2a^2+a+1)x+a^2+1$
• $y^2=(2a^2+a)x^6+a^2x^5+2a^2x^4+(a^2+a)x^3+(2a^2+2a+1)x^2+(2a^2+2a+2)x+a^2+a+2$
• $y^2=(2a^2+2a+2)x^6+(2a^2+a)x^5+(a+1)x^4+x^3+(a+1)x^2+(2a^2+a)x+2a^2+2a+2$
• $y^2=(a^2+2a+1)x^5+(2a^2+a)x^4+(a+2)x^3+(2a^2+a)x^2+(a^2+2a+1)x$
• $y^2=(2a^2+1)x^6+(a^2+2a+1)x^5+(a+1)x^4+(2a+1)x^3+(a^2+2a)x^2+(2a^2+2a+2)x+2a^2+2$
• $y^2=(a^2+2a+2)x^6+ax^5+(2a^2+a+1)x^4+(a^2+1)x^3+(2a^2+a+1)x^2+ax+a^2+2a+2$
• $y^2=(2a^2+a+2)x^6+(2a^2+2)x^5+(2a^2+2)x^4+ax^3+(2a+2)x^2+(a^2+a)x+a^2+1$
• $y^2=(2a^2+2a)x^6+(2a^2+1)x^5+(a^2+2a)x^4+(2a^2+2a)x^3+(2a+2)x^2+(2a^2+a)x+2a^2+a$
• $y^2=(2a^2+2a+2)x^6+(a+1)x^5+(a^2+a+2)x^4+(2a+1)x^3+(a^2+a+2)x^2+(a+1)x+2a^2+2a+2$
• and

• $y^2=(a+1)x^6+(2a^2+2a)x^5+(2a^2+a+2)x^4+2x^3+(2a^2+a+1)x^2+(2a^2+2a+2)x+2a^2$
• $y^2=(2a^2+2a+2)x^6+(a^2+2a)x^5+(2a+2)x^4+(2a^2+1)x^3+(2a+2)x^2+(a^2+2a)x+2a^2+2a+2$
• $y^2=2a^2x^6+(a^2+2a)x^5+(a^2+2a+1)x^4+(2a^2+2a)x^3+x^2+(2a^2+a+1)x+2$
• $y^2=(2a^2+2a)x^6+(a^2+a+1)x^5+(a^2+1)x^4+(2a^2+2a)x^3+2x^2+(2a+1)x+2a^2+1$
• $y^2=(a^2+1)x^6+ax^5+(a^2+2a+2)x^4+(a^2+a+1)x^3+(a^2+2a+2)x^2+ax+a^2+1$
• $y^2=2x^6+ax^5+(a^2+2a)x^4+a^2x^3+(2a^2+1)x^2+(a^2+a+1)x+2a$
• $y^2=(a^2+2a+2)x^6+a^2x^5+(2a^2+2)x^4+(a+1)x^3+(2a^2+2)x^2+a^2x+a^2+2a+2$
• $y^2=(2a^2+2a)x^6+a^2x^5+(2a^2+2a+1)x^4+(2a^2+a)x^3+(a^2+2a+1)x^2+(2a+1)x+2a^2+a+2$
• $y^2=(2a^2+a)x^6+(2a^2+a+1)x^5+(2a^2+1)x^4+(2a^2+a)x^3+(2a^2+a+2)x^2+(a^2+a+1)x+a+1$
• $y^2=(a+2)x^6+(2a^2+2a+2)x^5+(2a^2+2)x^4+2ax^3+a^2x^2+(a^2+2a+2)x+a^2+a+2$
• $y^2=(a^2+2a)x^5+(2a^2+2)x^4+(a^2+a+1)x^3+(a^2+2a)x^2+(2a+2)x$
• $y^2=2x^6+(a^2+2a+2)x^5+(2a^2+2a)x^4+(a^2+a+1)x^3+(2a^2+a+2)x^2+2x+2a^2+2a+2$
• $y^2=ax^6+ax^5+(a^2+2a+2)x^4+(a^2+2a)x^3+(a^2+2a+2)x^2+ax+a$
• $y^2=(2a^2+a)x^6+(a^2+a)x^5+(a^2+2a+2)x^4+(2a^2+2)x^3+(a^2+2a+2)x^2+(a^2+a)x+2a^2+a$
• $y^2=(a+1)x^6+(2a^2+a+2)x^5+(2a^2+2a+1)x^4+(2a+1)x^3+(a^2+a+1)x^2+(a^2+1)x+a^2+2a+2$
• $y^2=(a^2+a)x^5+(2a+2)x^4+(2a+1)x^3+(a^2+a)x^2+(a^2+2)x$
• $y^2=(a^2+a)x^6+(2a^2+2a+1)x^5+(2a+1)x^4+(a^2+a+2)x^3+(2a+1)x^2+(2a^2+2a+1)x+a^2+a$
• $y^2=(2a^2+2a+2)x^6+(a^2+a+2)x^4+(2a^2+2a+1)x^3+(2a+2)x^2+(2a^2+2)x+a+1$
• $y^2=ax^5+(2a^2+2a+1)x^4+(a^2+1)x^3+(2a^2+2a+1)x^2+ax$
• $y^2=(a^2+1)x^6+(2a^2+a+1)x^5+(2a^2+a+2)x^4+(a^2+2)x^3+x^2+(2a^2+2a)x+2$
• $y^2=2a^2x^6+(2a^2+2a+1)x^5+(2a^2+a+1)x^4+(a^2+2a+2)x^3+(2a^2+a+2)x^2+a^2+a+2$
• $y^2=(2a^2+a)x^6+(2a^2+a+1)x^5+(2a^2+2a+1)x^4+x^3+(2a^2+2a+1)x^2+(2a^2+a+1)x+2a^2+a$
• $y^2=(2a^2+a+2)x^6+(2a^2+1)x^5+(a^2+2a+1)x^4+(2a^2+a+2)x^3+(a^2+2a+1)x^2+(2a^2+1)x+2a^2+a+2$
• $y^2=(a^2+a+2)x^6+(a^2+2)x^5+(a^2+2)x^4+ax^3+(2a^2+2)x^2+(a^2+2a)x+2a^2+1$
• $y^2=(2a^2+a)x^6+2x^5+(2a^2+a)x^4+x^3+(2a^2+a)x^2+2x+2a^2+a$
• $y^2=ax^6+2a^2x^5+(2a^2+a+1)x^4+(2a+2)x^3+(a^2+2a+1)x^2+(a^2+a+2)x+a^2+1$
• $y^2=a^2x^5+(a^2+2)x^4+(2a^2+a+1)x^3+a^2x^2+(2a^2+2)x$
• $y^2=(2a^2+a+2)x^6+(a^2+a+1)x^5+(2a^2+2a+1)x^4+2ax^3+(2a^2+2a+1)x^2+(a^2+a+1)x+2a^2+a+2$
• $y^2=(2a^2+a+2)x^6+(2a^2+a+2)x^5+(a+1)x^4+2ax^3+(a+1)x^2+(2a^2+a+2)x+2a^2+a+2$
• $y^2=(a^2+2a+2)x^6+(a^2+a)x^5+(2a^2+1)x^4+2a^2x^3+(a^2+a+2)x^2+(2a^2+2a+2)x+2a^2+1$
• $y^2=2x^6+(2a+2)x^5+(2a+2)x^4+ax^3+(a^2+2)x^2+a^2x+a+1$
• $y^2=(a+1)x^6+(a^2+1)x^5+(2a^2+2a+2)x^4+(a^2+a+1)x^3+(2a^2+a+1)x^2+(2a^2+a)x+2a^2$
• $y^2=2a^2x^6+(2a^2+a)x^5+(a^2+1)x^4+(2a^2+2a+1)x^3+(a^2+1)x^2+(2a^2+a)x+2a^2$
• $y^2=(2a^2+2a)x^6+(2a^2+a)x^5+(a^2+a)x^4+(a^2+2a)x^3+(a^2+a)x^2+(2a^2+a)x+2a^2+2a$

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.ai $\times$ 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.ai : \(\Q(\sqrt{-11}) \).
• 1.27.ae : \(\Q(\sqrt{-23}) \).

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
2.27.ae_w	$2$	2.729.bc_bpm
2.27.e_w	$2$	2.729.bc_bpm
2.27.m_di	$2$	2.729.bc_bpm