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

• Label: 2.11.ai_bj
• 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:	$1 - 8 x + 35 x^{2} - 88 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.167863583547$, $\pm0.388927483238$
Angle rank:	$2$ (numerical)
Number field:	4.0.62352.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$61$	$15433$	$1859524$	$215583577$	$25923018781$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$128$	$1396$	$14724$	$160964$	$1772894$	$19500380$	$214400260$	$2357944300$	$25936950368$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=2 x^6+6 x^5+2 x^4+4 x^3+9 x^2+2 x+7$
• $y^2=7 x^6+8 x^5+2 x^3+10 x^2+10 x+9$

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 4.0.62352.1.

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.i_bj	$2$	2.121.g_ch