Abelian variety isogeny class 2.64.ax_jl over $\F_{2^{6}}$

• Label: 2.64.ax_jl
• Base field: $\F_{2^{6}}$
• 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:	$\F_{2^{6}}$
Dimension:	$2$
L-polynomial:	$1 - 23 x + 245 x^{2} - 1472 x^{3} + 4096 x^{4}$
Frobenius angles:	$\pm0.0870708726221$, $\pm0.342563904315$
Angle rank:	$2$ (numerical)
Number field:	4.0.13786305.2
Galois group:	$D_{4}$
Jacobians:	$24$
Isomorphism classes:	24

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2847$	$16617939$	$68803699536$	$281466424902195$	$1152863693810239827$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$4058$	$262467$	$16776706$	$1073687982$	$68718997271$	$4398046603818$	$281475019148386$	$18014398977602043$	$1152921507076730978$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2^{6}}$

The endomorphism algebra of this simple isogeny class is 4.0.13786305.2.

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
2.64.x_jl	$2$	(not in LMFDB)