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

• Label: 2.11.ae_n
• 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:	$1 - 4 x + 13 x^{2} - 44 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.179559488784$, $\pm0.577818478944$
Angle rank:	$2$ (numerical)
Number field:	4.0.88049.1
Galois group:	$D_{4}$
Jacobians:	$14$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$87$	$15921$	$1720512$	$214631001$	$26182476087$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$132$	$1292$	$14660$	$162568$	$1774422$	$19484536$	$214375684$	$2357991572$	$25936919652$

Jacobians and polarizations

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

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

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 4.0.88049.1.

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_n	$2$	2.121.k_ch