# Properties

 Label 2.11.ai_bi Base field $\F_{11}$ Dimension $2$ $p$-rank $2$ Ordinary Yes 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: $( 1 - 6 x + 11 x^{2} )( 1 - 2 x + 11 x^{2} )$ Frobenius angles: $\pm0.140218899004$, $\pm0.402508885479$ Angle rank: $2$ (numerical) Jacobians: 8

This isogeny class is not 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 8 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 60 15120 1826460 213857280 25884541500 3141328554000 380066253715740 45960389686149120 5559987016732558140 672744193373765898000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 126 1372 14606 160724 1773198 19503404 214408606 2357977252 25937200926

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The isogeny class factors as 1.11.ag $\times$ 1.11.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ae_k $2$ 2.121.e_ak 2.11.e_k $2$ 2.121.e_ak 2.11.i_bi $2$ 2.121.e_ak