# Properties

 Label 2.16.ak_cb 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: $( 1 - 7 x + 16 x^{2} )( 1 - 3 x + 16 x^{2} )$ Frobenius angles: $\pm0.160861246510$, $\pm0.377642706461$ Angle rank: $2$ (numerical) Jacobians: 16

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 16 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 140 67200 17235260 4308595200 1099248363500 281519358019200 72069543068626460 18447655167170611200 4722383634715326216140 1208923287933055002000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 263 4207 65743 1048327 16779863 268479967 4295179423 68719726327 1099509325223

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ah $\times$ 1.16.ad 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.ae_l $2$ 2.256.g_er 2.16.e_l $2$ 2.256.g_er 2.16.k_cb $2$ 2.256.g_er