# Properties

 Label 2.11.ag_w Base field $\F_{11}$ Dimension $2$ $p$-rank $1$ Ordinary No Supersingular No Simple No Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{11}$ Dimension: $2$ L-polynomial: $( 1 - 6 x + 11 x^{2} )( 1 + 11 x^{2} )$ Frobenius angles: $\pm0.140218899004$, $\pm0.5$ Angle rank: $1$ (numerical) Jacobians: 16

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class contains the Jacobians of 16 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 72 15552 1750248 211507200 26014085352 3147295953600 379921632491592 45949275913420800 5560073041789601928 672760876491684395712

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 130 1314 14446 161526 1776562 19495986 214356766 2358013734 25937844130

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The isogeny class factors as 1.11.ag $\times$ 1.11.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{11}$
 The base change of $A$ to $\F_{11^{2}}$ is 1.121.ao $\times$ 1.121.w. The endomorphism algebra for each factor is: 1.121.ao : $$\Q(\sqrt{-2})$$. 1.121.w : the quaternion algebra over $$\Q$$ ramified at $11$ and $\infty$.
All geometric endomorphisms are defined over $\F_{11^{2}}$.

## 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_w $2$ 2.121.i_aco