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

• Label: 2.11.ae_u
• 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:	$1 - 4 x + 20 x^{2} - 44 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.265956980117$, $\pm0.521586215300$
Angle rank:	$2$ (numerical)
Number field:	4.0.610560.3
Galois group:	$D_{4}$
Jacobians:	$4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$94$	$17860$	$1831966$	$214391440$	$26046727054$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$146$	$1376$	$14646$	$161728$	$1773218$	$19475128$	$214306846$	$2357971496$	$25937830226$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

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

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 4.0.610560.3.

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_u	$2$	2.121.y_le