Abelian variety isogeny class 2.11.a_aw over $\F_{11}$

• Label: 2.11.a_aw
• Base field: $\F_{11}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$( 1 - 11 x^{2} )^{2}$
	$1 - 22 x^{2} + 121 x^{4}$
Frobenius angles:	$0$, $0$, $1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{11}) \)
Galois group:	$C_2$
Jacobians:	$1$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

This isogeny class is simple but not geometrically simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$100$	$10000$	$1768900$	$207360000$	$25937102500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$78$	$1332$	$14158$	$161052$	$1766238$	$19487172$	$214300318$	$2357947692$	$25936780398$

Jacobians and polarizations

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

• $y^2=x^6+x^5+10 x^4+6 x^3+x^2+x+10$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{2}}$.

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{11}) \) ramified at both real infinite places.

Endomorphism algebra over $\overline{\F}_{11}$

The base change of $A$ to $\F_{11^{2}}$ is 1.121.aw^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $11$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.11.a_l	$3$	(not in LMFDB)
2.11.a_w	$4$	(not in LMFDB)
2.11.a_l	$6$	(not in LMFDB)