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

• Label: 2.64.abb_lt
• 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:	$1 - 27 x + 305 x^{2} - 1728 x^{3} + 4096 x^{4}$
Frobenius angles:	$\pm0.0514697705289$, $\pm0.252940519903$
Angle rank:	$2$ (numerical)
Number field:	4.0.381465.2
Galois group:	$D_{4}$
Jacobians:	$6$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2647$	$16297579$	$68676859696$	$281502622378035$	$1152920316123871327$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$3978$	$261983$	$16778866$	$1073740718$	$68719151391$	$4398041577422$	$281474934064546$	$18014398305146687$	$1152921504904070778$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.381465.2.

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.bb_lt	$2$	(not in LMFDB)