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

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

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$1$
L-polynomial:	$1 + 4 x^{2}$
Frobenius angles:	$\pm0.5$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-1}) \)
Galois group:	$C_2$
Jacobians:	$1$

This isogeny class is simple and geometrically simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5$	$25$	$65$	$225$	$1025$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$25$	$65$	$225$	$1025$	$4225$	$16385$	$65025$	$262145$	$1050625$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

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

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

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 1.16.i and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

Subfield	Primitive Model
$\F_{2}$	1.2.ac
$\F_{2}$	1.2.c

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
1.4.ae	$4$	1.256.abg
1.4.e	$4$	1.256.abg
1.4.ac	$12$	(not in LMFDB)