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

• Label: 2.11.ag_y
• 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:	$1 - 6 x + 24 x^{2} - 66 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.175918430288$, $\pm0.482992569757$
Angle rank:	$2$ (numerical)
Number field:	4.0.417088.1
Galois group:	$D_{4}$
Jacobians:	$4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$74$	$16132$	$1798274$	$213006928$	$26023590434$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$134$	$1350$	$14550$	$161586$	$1776710$	$19496994$	$214340638$	$2357877654$	$25937467574$

Jacobians and polarizations

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

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

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 4.0.417088.1.

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.g_y	$2$	2.121.m_ba