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

• Label: 2.11.af_m
• 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:	$1 - 5 x + 12 x^{2} - 55 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.0559264965648$, $\pm0.574142933607$
Angle rank:	$2$ (numerical)
Number field:	4.0.236600.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	4
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$74$	$14356$	$1630664$	$209597600$	$25973605654$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$121$	$1222$	$14313$	$161277$	$1770706$	$19475407$	$214362513$	$2358032722$	$25937262681$

Jacobians and polarizations

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

• $y^2=2 x^6+9 x^5+4 x^4+2 x^3+8 x^2+6 x+1$
• $y^2=4 x^6+9 x^5+3 x^4+2 x^3+x^2+9 x+8$

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 4.0.236600.1.

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.f_m	$2$	2.121.ab_agi