Abelian variety isogeny class 2.4.ad_i over $\F_{2^{2}}$

• Label: 2.4.ad_i
• Base field: $\F_{2^{2}}$
• 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:	$\F_{2^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 3 x + 4 x^{2} )( 1 + 4 x^{2} )$
	$1 - 3 x + 8 x^{2} - 12 x^{3} + 16 x^{4}$
Frobenius angles:	$\pm0.230053456163$, $\pm0.5$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10$	$400$	$4810$	$64800$	$1109050$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$24$	$74$	$256$	$1082$	$4272$	$16298$	$64576$	$261146$	$1049424$

Jacobians and polarizations

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

• $y^2+xy=x^5+x^3+(a+1)x^2+x$

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.4.ad $\times$ 1.4.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.4.ad : \(\Q(\sqrt{-7}) \).
• 1.4.a : \(\Q(\sqrt{-1}) \).

Endomorphism algebra over $\overline{\F}_{2^{2}}$

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i. The endomorphism algebra for each factor is:

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

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
2.4.d_i	$2$	2.16.h_y
2.4.ah_u	$4$	2.256.ab_asm
2.4.ab_ae	$4$	2.256.ab_asm