# Properties

 Label 2.16.ah_bn 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: $1 - 7 x + 39 x^{2} - 112 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.242342613760$, $\pm0.451721988494$ Angle rank: $2$ (numerical) Number field: 4.0.840105.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 177 73455 17363700 4300716795 1099724316387 281579135598000 72060185239793277 18445994569161355155 4722300860256667914300 1208925394929305799426375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 286 4237 65626 1048780 16783423 268445110 4294792786 68718521797 1099511241526

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.840105.1.
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.h_bn $2$ 2.256.bd_rx