Abelian variety isogeny class 2.16.ah_y over $\F_{2^{4}}$

• Label: 2.16.ah_y
• Base field: $\F_{2^{4}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 4 x )^{2}( 1 + x + 16 x^{2} )$
	$1 - 7 x + 24 x^{2} - 112 x^{3} + 256 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.539893087675$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$162$	$64800$	$16074450$	$4232347200$	$1098623120562$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$256$	$3922$	$64576$	$1047730$	$16775008$	$268377490$	$4294765696$	$68719426162$	$1099510185376$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2+xy=x^5+x^3+(a^3+a+1)x^2+x$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2^{4}}$

The isogeny class factors as 1.16.ai $\times$ 1.16.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.16.b : \(\Q(\sqrt{-7}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.16.aj_bo	$2$	2.256.ab_asm
2.16.h_y	$2$	2.256.ab_asm
2.16.j_bo	$2$	2.256.ab_asm
2.16.f_bk	$3$	(not in LMFDB)