Abelian variety isogeny class 2.32.ao_dz over $\F_{2^{5}}$

• Label: 2.32.ao_dz
• Base field: $\F_{2^{5}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{5}}$
Dimension:	$2$
L-polynomial:	$1 - 14 x + 103 x^{2} - 448 x^{3} + 1024 x^{4}$
Frobenius angles:	$\pm0.144855693610$, $\pm0.389840297544$
Angle rank:	$2$ (numerical)
Number field:	4.0.4481600.2
Galois group:	$D_{4}$
Jacobians:	$50$
Isomorphism classes:	100

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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$666$	$1058940$	$1081583334$	$1099645653600$	$1125767884980906$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$19$	$1035$	$33007$	$1048703$	$33550499$	$1073770155$	$34360352927$	$1099515422143$	$35184378630739$	$1125899853921675$

Jacobians and polarizations

This isogeny class contains the Jacobians of 50 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{5}}$.

Endomorphism algebra over $\F_{2^{5}}$

The endomorphism algebra of this simple isogeny class is 4.0.4481600.2.

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
2.32.o_dz	$2$	2.1024.k_ej