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

• Label: 2.4.ab_c
• 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:	$( 1 - 3 x + 4 x^{2} )( 1 + 2 x + 4 x^{2} )$
	$1 - x + 2 x^{2} - 4 x^{3} + 16 x^{4}$
Frobenius angles:	$\pm0.230053456163$, $\pm0.666666666667$
Angle rank:	$1$ (numerical)
Jacobians:	$2$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14$	$336$	$3626$	$78624$	$1143674$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$20$	$58$	$304$	$1114$	$4016$	$16426$	$65344$	$260122$	$1048400$

Jacobians and polarizations

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

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

where $a$ is a root of the Conway polynomial.

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.ad $\times$ 1.4.c 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.c : \(\Q(\sqrt{-3}) \).

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

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

• 1.64.aq : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.64.j : \(\Q(\sqrt{-7}) \).

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.af_o	$2$	2.16.d_bc
2.4.b_c	$2$	2.16.d_bc
2.4.f_o	$2$	2.16.d_bc
2.4.ah_u	$3$	2.64.ah_aq