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

• Label: 2.4.a_ab
• Base field: $\F_{2^{2}}$
• 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:	$\F_{2^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 3 x + 4 x^{2} )( 1 + 3 x + 4 x^{2} )$
	$1 - x^{2} + 16 x^{4}$
Frobenius angles:	$\pm0.230053456163$, $\pm0.769946543837$
Angle rank:	$1$ (numerical)
Jacobians:	$2$
Isomorphism classes:	20

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$256$	$4144$	$82944$	$1047376$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$15$	$65$	$319$	$1025$	$4191$	$16385$	$64639$	$262145$	$1046175$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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.d 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.d : \(\Q(\sqrt{-7}) \).

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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
2.4.ag_r	$2$	2.16.ac_bh
2.4.g_r	$2$	2.16.ac_bh
2.4.a_b	$4$	2.256.ck_cer