# Properties

 Label 2.16.ak_cf 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: $( 1 - 5 x + 16 x^{2} )^{2}$ Frobenius angles: $\pm0.285098958592$, $\pm0.285098958592$ Angle rank: $1$ (numerical) Jacobians: 6

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 6 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 144 69696 17740944 4356000000 1100510098704 281306156129856 72040005072846864 18446028573456000000 4722381292753126898064 1208929935151727208484416

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 271 4327 66463 1049527 16767151 268369927 4294800703 68719692247 1099515370831

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-39})$$$)$
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.a_h $2$ 2.256.o_vp 2.16.k_cf $2$ 2.256.o_vp 2.16.f_j $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.a_h $2$ 2.256.o_vp 2.16.k_cf $2$ 2.256.o_vp 2.16.f_j $3$ (not in LMFDB) 2.16.a_ah $4$ (not in LMFDB) 2.16.af_j $6$ (not in LMFDB)