Abelian variety isogeny class 2.128.abn_yd over $\F_{2^{7}}$

• Label: 2.128.abn_yd
• Base field: $\F_{2^{7}}$
• 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:	$\F_{2^{7}}$
Dimension:	$2$
L-polynomial:	$1 - 39 x + 627 x^{2} - 4992 x^{3} + 16384 x^{4}$
Frobenius angles:	$\pm0.0277667321603$, $\pm0.240740084074$
Angle rank:	$2$ (numerical)
Number field:	4.0.1282753.2
Galois group:	$D_{4}$
Jacobians:	$7$
Isomorphism classes:	7

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11981$	$264097183$	$4396082561732$	$72058064139940459$	$1180590292716679643051$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$16118$	$2096217$	$268437210$	$34359699720$	$4398043606679$	$562949888468022$	$72057593104167730$	$9223372027298171817$	$1180591620653510684438$

Jacobians and polarizations

This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{7}}$.

Endomorphism algebra over $\F_{2^{7}}$

The endomorphism algebra of this simple isogeny class is 4.0.1282753.2.

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
2.128.bn_yd	$2$	(not in LMFDB)