Abelian variety isogeny class 1.4.ac over $\F_{2^{2}}$

• Label: 1.4.ac
• Base field: $\F_{2^{2}}$
• Dimension: $1$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$1$
L-polynomial:	$1 - 2 x + 4 x^{2}$
Frobenius angles:	$\pm0.333333333333$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	$2$
Isomorphism classes:	2

This isogeny class is simple and geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3$	$21$	$81$	$273$	$993$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$21$	$81$	$273$	$993$	$3969$	$16257$	$65793$	$263169$	$1049601$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):

• $y^2+a y=x^3$
• $y^2+(a+1) y=x^3$

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 endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 1.64.q and its endomorphism algebra is 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
1.4.c	$2$	1.16.e
1.4.e	$3$	1.64.q
1.4.ae	$6$	(not in LMFDB)