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

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

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$1 - 10 x + 45 x^{2} - 110 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.0820279942768$, $\pm0.318205720493$
Angle rank:	$2$ (numerical)
Number field:	4.0.5696.1
Galois group:	$D_{4}$
Jacobians:	$1$
Isomorphism classes:	1
Cyclic group of points:	yes

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$47$	$13489$	$1797092$	$214866281$	$25865323927$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$112$	$1352$	$14676$	$160602$	$1768942$	$19482542$	$214376868$	$2358102152$	$25937967552$

Jacobians and polarizations

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

• $y^2=6 x^6+6 x^5+10 x^4+3 x^3+8 x^2+2$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is 4.0.5696.1.

Base change

This is a primitive isogeny class.

Twists

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

Twist	Extension degree	Common base change
2.11.k_bt	$2$	2.121.ak_cp