# Properties

 Label 2.16.ak_bz 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: $1 - 10 x + 51 x^{2} - 160 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.118775077357$, $\pm0.396715540983$ Angle rank: $2$ (numerical) Number field: 4.0.281664.1 Galois group: $D_{4}$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 138 65964 16984350 4283438304 1097992070778 281509656727500 72071910946137198 18447781339996408704 4722393761144452492650 1208925442263001091952684

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 259 4147 65359 1047127 16779283 268488787 4295208799 68719873687 1099511284579

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.281664.1.
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.k_bz $2$ 2.256.c_adj