# Properties

 Label 2.16.ai_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: $( 1 - 7 x + 16 x^{2} )( 1 - x + 16 x^{2} )$ Frobenius angles: $\pm0.160861246510$, $\pm0.460106912325$ Angle rank: $2$ (numerical) Jacobians: 34

This isogeny class is not 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 34 curves, and hence is principally polarizable:

• $y^2+(x^2+x)y=(a^2+1)x^5+(a^3+a^2+a+1)x^3+(a^3+a^2+1)x^2+(a^2+a+1)x$
• $y^2+(x^2+x+a^3+a+1)y=(a^2+1)x^5+(a^3+a^2+a)x^4+(a^3+a^2+a)x^2+(a^3+a^2+a+1)x+a^2+a$
• $y^2+(x^2+x)y=ax^5+(a^3+a)x^3+(a^3+a^2+a)x^2+(a^2+a)x$
• $y^2+(x^2+x+a^3+1)y=ax^5+(a^3+a+1)x^4+(a^3+a+1)x^2+(a^3+a)x+a^2+a+1$
• $y^2+(x^2+x+a^3+a)y=(a^2+a)x^5+(a^3+a+1)x^4+ax^3+(a^3+a+1)x^2+(a^3+a^2+a+1)x+a^3+1$
• $y^2+(x^2+x+a^3+a^2+1)y=(a+1)x^5+(a^3+a^2+1)x^4+(a^2+a)x^3+(a^3+a^2+1)x^2+a^3x+a^2+a$
• $y^2+(x^2+x+a^3+a)y=(a^2+a+1)x^5+(a^3+a)x^4+(a^2+a+1)x^3+(a^3+1)x^2+(a^3+a^2+a)x+a$
• $y^2+(x^2+x)y=a^2x^5+a^3x^3+(a^2+a+1)x^2+(a^3+a+1)x$
• $y^2+(x^2+x+a^3+a^2+1)y=a^2x^5+(a^3+a+1)x^4+(a^3+a+1)x^2+a^3x+a^3+1$
• $y^2+(x^2+x+a^3+a^2+1)y=(a+1)x^5+(a^3+a^2+1)x^4+(a+1)x^3+a^3x^2+a^2x+a^2+a$
• $y^2+(x^2+x+a^3+a+1)y=ax^5+(a^3+a+1)x^4+ax^3+(a^3+a^2+a+1)x^2+(a^2+1)x+a^2+a$
• $y^2+(x^2+x+a^3+a^2+a)y=x^5+x^3+(a^2+a+1)x^2+(a^3+a^2)x+a^3+a^2$
• $y^2+(x^2+x+a^3+a^2+1)y=(a^2+a)x^5+(a^3+a^2+1)x^4+(a^2+a)x^3+a^3x^2+(a^2+a+1)x+a^2+a$
• $y^2+(x^2+x+a^3)y=(a^2+a+1)x^5+(a^3+a+1)x^4+a^2x^3+(a^3+a+1)x^2+(a^3+a)x+a^2+a$
• $y^2+(x^2+x+a^3+a^2+a)y=(a^2+1)x^5+a^3x^4+(a^2+a+1)x^3+a^3x^2+(a^3+a^2)x+a^3$
• $y^2+(x^2+x)y=(a^2+a)x^5+(a^3+a^2)x^4+(a^3+1)x^3+(a^2+a)x^2+(a^2+1)x$
• $y^2+(x^2+x)y=x^5+(a^3+a^2+1)x^3+(a^3+1)x^2+(a^2+1)x$
• $y^2+(x^2+x)y=(a+1)x^5+(a^3+a^2)x^3+(a^3+1)x^2+(a^2+a)x$
• $y^2+(x^2+x+a^3+a^2+a)y=(a+1)x^5+(a^3+a^2+a)x^4+(a^3+a^2+a)x^2+(a^3+a^2)x+a^3+1$
• $y^2+(x^2+x+a^3+a^2+1)y=(a^2+a)x^5+(a^2+a)x^3+a^2x^2+(a^2+a)x+1$
• and 14 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 160 69120 16948960 4280186880 1100006644000 281712053952000 72072475250892640 18446790522338795520 4722339817060746727840 1208925767910734900928000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 271 4137 65311 1049049 16791343 268490889 4294978111 68719088697 1099511580751

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ah $\times$ 1.16.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ag_z $2$ 2.256.o_ap 2.16.g_z $2$ 2.256.o_ap 2.16.i_bn $2$ 2.256.o_ap