# Properties

 Label 2.16.ai_bj 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: $1 - 8 x + 35 x^{2} - 128 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.100372839389$, $\pm0.484299017784$ Angle rank: $2$ (numerical) Number field: 4.0.66417.2 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 156 66768 16554564 4252053312 1099002013356 281649882685392 72065800325865684 18446682900457813248 4722385389546620728764 1208930156277080585728848

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 263 4041 64879 1048089 16787639 268466025 4294953055 68719751865 1099515571943

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.66417.2.
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_bj $2$ 2.256.g_alz