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

• Label: 2.11.ae_s
• 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:	$1 - 4 x + 18 x^{2} - 44 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.240490560995$, $\pm0.539857825313$
Angle rank:	$2$ (numerical)
Number field:	4.0.14336.1
Galois group:	$D_{4}$
Jacobians:	$14$
Isomorphism classes:	18

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$92$	$17296$	$1799612$	$214747136$	$26117784412$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$142$	$1352$	$14670$	$162168$	$1773982$	$19475576$	$214314654$	$2357950952$	$25937430702$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

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

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 4.0.14336.1.

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.e_s	$2$	2.121.u_ig