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

• Label: 2.4.ab_g
• 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:	$( 1 - 2 x + 4 x^{2} )( 1 + x + 4 x^{2} )$
	$1 - x + 6 x^{2} - 4 x^{3} + 16 x^{4}$
Frobenius angles:	$\pm0.333333333333$, $\pm0.580430623255$
Angle rank:	$1$ (numerical)
Jacobians:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18$	$504$	$4374$	$65520$	$1078398$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$28$	$70$	$256$	$1054$	$3976$	$16006$	$66016$	$263950$	$1047928$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{6}}$.

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

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

• 1.4.ac : \(\Q(\sqrt{-3}) \).
• 1.4.b : \(\Q(\sqrt{-15}) \).

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

The base change of $A$ to $\F_{2^{6}}$ is 1.64.al $\times$ 1.64.q. The endomorphism algebra for each factor is:

• 1.64.al : \(\Q(\sqrt{-15}) \).
• 1.64.q : 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.ad_k	$2$	2.16.l_ci
2.4.b_g	$2$	2.16.l_ci
2.4.d_k	$2$	2.16.l_ci
2.4.f_m	$3$	2.64.f_abw