# Properties

 Label 2.11.af_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: $1 - 5 x + 18 x^{2} - 55 x^{3} + 121 x^{4}$ Frobenius angles: $\pm0.170748385587$, $\pm0.533728959213$ Angle rank: $2$ (numerical) Number field: 4.0.8405.1 Galois group: $D_{4}$ Jacobians: 10

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 10 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 80 16000 1747520 213056000 26119222000 3146934016000 379774285897520 45948141875456000 5560077704924386880 672749013766810000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 133 1312 14553 162177 1776358 19488427 214351473 2358015712 25937386773

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The endomorphism algebra of this simple isogeny class is 4.0.8405.1.
All geometric endomorphisms are defined over $\F_{11}$.

## 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.f_s $2$ 2.121.l_q