# Properties

 Label 2.16.ai_bl 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: $1 - 8 x + 37 x^{2} - 128 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.132522875726$, $\pm0.472776187397$ Angle rank: $2$ (numerical) Number field: 4.0.1287440.3 Galois group: $D_{4}$ Jacobians: 16

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 16 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 158 67940 16751318 4266632000 1099672258158 281704737134660 72071122163901158 18446860746305312000 4722371034278166721598 1208928535686932465038500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 267 4089 65103 1048729 16790907 268485849 4294994463 68719542969 1099514098027

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.1287440.3.
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.i_bl $2$ 2.256.k_agl