# Properties

 Label 2.16.al_cj Base Field $\F_{2^{4}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $2$ L-polynomial: $1 - 11 x + 61 x^{2} - 176 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.189901625224$, $\pm0.315486115946$ Angle rank: $2$ (numerical) Number field: 4.0.22625.1 Galois group: $D_{4}$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

• $y^2+(x^3+a^2x+a^2)y=(a^3+a^2+1)x^6+(a^2+a)x^5+(a^2+a)x^4+(a+1)x^3+a^2x^2+(a^3+a^2+1)x+a$
• $y^2+(x^3+(a+1)x+a+1)y=(a^3+a^2)x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+(a^2+1)x^3+(a^3+1)x^2+(a^3+a^2+a)x+a^3+a^2+a$
• $y^2+(x^3+ax+a)y=(a^3+a)x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+a^2x^3+(a^3+a^2+a)x^2+(a^3+1)x+a^3+1$
• $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a^2)x^6+(a^2+a)x^5+(a^2+a)x^4+ax^3+(a^3+a+1)x^2+(a^3+a+1)x+a^3+a^2+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 131 66155 17419856 4342480355 1101116001591 281469684850880 72055110843243491 18446752332853960355 4722376017962412943376 1208925667133961439921875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 258 4251 66258 1050106 16776903 268426206 4294969218 68719615491 1099511489098

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.22625.1.
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## 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 2.16.l_cj $2$ 2.256.b_nx