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

• Label: 2.16.am_cp
• Base field: $\F_{2^{4}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• 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:	$( 1 - 7 x + 16 x^{2} )( 1 - 5 x + 16 x^{2} )$
	$1 - 12 x + 67 x^{2} - 192 x^{3} + 256 x^{4}$
Frobenius angles:	$\pm0.160861246510$, $\pm0.285098958592$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Isomorphism classes:	32

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$120$	$63360$	$17227080$	$4340160000$	$1101766863000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$247$	$4205$	$66223$	$1050725$	$16780327$	$268432925$	$4294965343$	$68719670165$	$1099512797527$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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.ah $\times$ 1.16.af 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.ah : \(\Q(\sqrt{-15}) \).
• 1.16.af : \(\Q(\sqrt{-39}) \).

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
2.16.ac_ad	$2$	2.256.ak_pd
2.16.c_ad	$2$	2.256.ak_pd
2.16.m_cp	$2$	2.256.ak_pd