# Properties

 Label 2.16.aj_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: $1 - 9 x + 45 x^{2} - 144 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.144241903460$, $\pm0.427458851042$ Angle rank: $2$ (numerical) Number field: 4.0.626545.2 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 149 67795 17004476 4282542355 1098935616329 281628131512000 72075181122945149 18447383044087046595 4722355873537529645036 1208924734672115319824875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 266 4151 65346 1048028 16786343 268500968 4295116066 68719322351 1099510641026

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.626545.2.
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.j_bt $2$ 2.256.j_acd