Abelian variety isogeny class 2.128.bp_zw over $\F_{2^{7}}$

• Label: 2.128.bp_zw
• Base field: $\F_{2^{7}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{7}}$
Dimension:	$2$
L-polynomial:	$1 + 41 x + 672 x^{2} + 5248 x^{3} + 16384 x^{4}$
Frobenius angles:	$\pm0.803192786549$, $\pm0.975707193065$
Angle rank:	$2$ (numerical)
Number field:	4.0.2312.1
Galois group:	$D_{4}$
Jacobians:	$1$

This isogeny class is simple and geometrically simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$22346$	$262967728$	$4402261596122$	$72056107443972704$	$1180590720467952179626$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$170$	$16048$	$2099162$	$268429920$	$34359712170$	$4398046284304$	$562949976468602$	$72057593473609920$	$9223372046123532362$	$1180591620599377727088$

Jacobians and polarizations

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

• $y^2+xy=x^5+x^2+x$

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 4.0.2312.1.

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
$\F_{2}$	2.2.ab_a

Twists

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

Twist	Extension degree	Common base change
2.128.abp_zw	$2$	(not in LMFDB)