Abelian variety isogeny class 1.128.q over $\F_{2^{7}}$

• Label: 1.128.q
• Base field: $\F_{2^{7}}$
• 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:	$\F_{2^{7}}$
Dimension:	$1$
L-polynomial:	$1 + 16 x + 128 x^{2}$
Frobenius angles:	$\pm0.750000000000$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$145$	$16385$	$2095105$	$268468225$	$34359476225$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$145$	$16385$	$2095105$	$268468225$	$34359476225$	$4398046511105$	$562949986975745$	$72057593501057025$	$9223372041149743105$	$1180591620717411303425$

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 $\F_{2^{28}}$.

Endomorphism algebra over $\F_{2^{7}}$

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

Endomorphism algebra over $\overline{\F}_{2^{7}}$

The base change of $A$ to $\F_{2^{28}}$ is the simple isogeny class 1.268435456.bwmi and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{14}}$

  The base change of $A$ to $\F_{2^{14}}$ is the simple isogeny class 1.16384.a and its endomorphism algebra is \(\Q(\sqrt{-1}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{7}}$.

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

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
1.128.aq	$2$	(not in LMFDB)
1.128.a	$8$	(not in LMFDB)