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

• Label: 2.32.ar_fd
• 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:	$1 - 17 x + 133 x^{2} - 544 x^{3} + 1024 x^{4}$
Frobenius angles:	$\pm0.135585099322$, $\pm0.298355681043$
Angle rank:	$2$ (numerical)
Number field:	4.0.307073.2
Galois group:	$D_{4}$
Jacobians:	$10$
Isomorphism classes:	10

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$597$	$1026243$	$1081570572$	$1101562052499$	$1126116167001537$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$1002$	$33007$	$1050530$	$33560876$	$1073744631$	$34359747872$	$1099512921250$	$35184387742807$	$1125900003224562$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.307073.2.

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.r_fd	$2$	2.1024.ax_bvt