# Properties

 Label 2.16.ai_bt 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: $1 - 8 x + 45 x^{2} - 128 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.245740075077$, $\pm0.408506512405$ Angle rank: $2$ (numerical) Number field: 4.0.263952.2 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 166 72708 17547694 4314783552 1098998890006 281464976664612 72059936860170334 18446559206942692608 4722319155192747303814 1208923287954487848426948

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 283 4281 65839 1048089 16776619 268444185 4294924255 68718788025 1099509325243

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.263952.2.
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_bt $2$ 2.256.ba_sv