# Properties

 Label 2.16.ah_bd 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: $1 - 7 x + 29 x^{2} - 112 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.123526934512$, $\pm0.516126302719$ Angle rank: $2$ (numerical) Number field: 4.0.241865.1 Galois group: $D_{4}$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 167 67635 16499600 4261613715 1100827971587 281700640728000 72062553043324367 18446789288232935235 4722416178984729976400 1208929539981090148360875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 266 4027 65026 1049830 16790663 268453930 4294977826 68720199907 1099515011426

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.241865.1.
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_bd $2$ 2.256.j_aih