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

• Label: 2.11.b_ac
• 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:	$1 + x - 2 x^{2} + 11 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.267574779296$, $\pm0.804786259760$
Angle rank:	$2$ (numerical)
Number field:	4.0.837401.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$132$	$14256$	$1826352$	$220625856$	$25871563212$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$117$	$1372$	$15065$	$160643$	$1772982$	$19475441$	$214326769$	$2357990212$	$25937275077$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=5 x^6+3 x^5+2 x^4+6 x^3+2 x^2+9 x$
• $y^2=x^6+5 x^5+5 x^4+x^3+5 x^2+9 x+7$
• $y^2=7 x^6+5 x^5+4 x^4+6 x^3+2 x^2+10 x+9$
• $y^2=5 x^6+7 x^5+9 x^4+6 x^3+9 x^2+5 x+6$
• $y^2=6 x^6+8 x^5+x^3+3 x^2+x+3$
• $y^2=4 x^6+x^5+8 x^4+x^3+5 x^2+2 x+8$
• $y^2=9 x^6+8 x^5+3 x^4+5 x^3+8 x^2+x+1$
• $y^2=5 x^6+9 x^5+2 x^4+9 x^3+10 x^2+7 x+4$
• $y^2=3 x^6+8 x^5+2 x^4+4 x^3+3 x^2+9 x$
• $y^2=x^6+x^5+4 x^4+3 x^3+3 x^2+8 x+9$
• $y^2=3 x^6+10 x^5+6 x^4+7 x^3+6 x^2+10 x+2$
• $y^2=5 x^6+3 x^5+8 x^4+5 x^3+5 x^2+2 x+6$

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 4.0.837401.1.

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.ab_ac	$2$	2.121.af_iq