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

• Label: 1.128.an
• Base field: $\F_{2^{7}}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{7}}$
Dimension:	$1$
L-polynomial:	$1 - 13 x + 128 x^{2}$
Frobenius angles:	$\pm0.305187096569$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-7}) \)
Galois group:	$C_2$
Jacobians:	$8$
Isomorphism classes:	8

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$116$	$16472$	$2099948$	$268460656$	$34359708196$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$116$	$16472$	$2099948$	$268460656$	$34359708196$	$4398042893384$	$562949910253084$	$72057593939809248$	$9223372041104766164$	$1180591620785220370232$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{7}}$.

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

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-7}) \).

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.b

Twists

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

Twist	Extension degree	Common base change
1.128.n	$2$	(not in LMFDB)