Abelian variety isogeny class 2.16.aq_ds over $\F_{2^{4}}$

• Label: 2.16.aq_ds
• Base field: $\F_{2^{4}}$
• 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:	$\F_{2^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 4 x )^{4}$
	$1 - 16 x + 96 x^{2} - 256 x^{3} + 256 x^{4}$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$81$	$50625$	$15752961$	$4228250625$	$1095222947841$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$193$	$3841$	$64513$	$1044481$	$16760833$	$268369921$	$4294705153$	$68718428161$	$1099507433473$

Jacobians and polarizations

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

• $y^2+y=x^5+a^3+a^2+a$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{2^{4}}$

The isogeny class factors as 1.16.ai^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 $\F_{2^{4}}$.

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

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.16.a_abg	$2$	2.256.acm_chc
2.16.q_ds	$2$	2.256.acm_chc
2.16.ae_a	$3$	(not in LMFDB)
2.16.i_bw	$3$	(not in LMFDB)