Abelian variety isogeny class 2.64.abg_ou over $\F_{2^{6}}$

• Label: 2.64.abg_ou
• Base field: $\F_{2^{6}}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{6}}$
Dimension:	$2$
L-polynomial:	$( 1 - 8 x )^{4}$
	$1 - 32 x + 384 x^{2} - 2048 x^{3} + 4096 x^{4}$
Frobenius angles:	$0$, $0$, $0$, $0$
Angle rank:	$0$ (numerical)
Jacobians:	$1$

This isogeny class is not 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:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2401$	$15752961$	$68184176641$	$281200199450625$	$1152780773560811521$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$33$	$3841$	$260097$	$16760833$	$1073610753$	$68718428161$	$4398038122497$	$281474909601793$	$18014397972611073$	$1152921500311879681$

Jacobians and polarizations

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

• $y^2+y=x^5+x^4+x^3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2^{6}}$

The isogeny class factors as 1.64.aq^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ 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 $\F_{2^{6}}$.

Subfield	Primitive Model
$\F_{2}$	2.2.a_ae
$\F_{2}$	2.2.a_c

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.64.a_aey	$2$	(not in LMFDB)
2.64.bg_ou	$2$	(not in LMFDB)
2.64.ai_a	$3$	(not in LMFDB)
2.64.q_hk	$3$	(not in LMFDB)