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

• Label: 2.64.abd_mz
• 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:	$1 - 29 x + 337 x^{2} - 1856 x^{3} + 4096 x^{4}$
Frobenius angles:	$\pm0.0696923787592$, $\pm0.184671871212$
Angle rank:	$2$ (numerical)
Number field:	4.0.23225.1
Galois group:	$D_{4}$
Jacobians:	$3$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2549$	$16107131$	$68552612276$	$281480634897299$	$1152957594379246749$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$3930$	$261507$	$16777554$	$1073775436$	$68719837791$	$4398048893484$	$281474985126114$	$18014398483114083$	$1152921503948987050$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.23225.1.

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.bd_mz	$2$	(not in LMFDB)