# Properties

 Label 2.16.ah_y Base Field $\F_{2^{4}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $2$ L-polynomial: $( 1 - 4 x )^{2}( 1 + x + 16 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.539893087675$ Angle rank: $1$ (numerical) Jacobians: 1

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2+xy=x^5+x^3+(a^3+a+1)x^2+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 162 64800 16074450 4232347200 1098623120562 281437900380000 72042035002383762 18445878221841820800 4722363007271186152050 1208924233678783668420000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 256 3922 64576 1047730 16775008 268377490 4294765696 68719426162 1099510185376

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 1.16.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.16.ai : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.16.b : $$\Q(\sqrt{-7})$$.
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.aj_bo $2$ 2.256.ab_asm 2.16.h_y $2$ 2.256.ab_asm 2.16.j_bo $2$ 2.256.ab_asm 2.16.f_bk $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.aj_bo $2$ 2.256.ab_asm 2.16.h_y $2$ 2.256.ab_asm 2.16.j_bo $2$ 2.256.ab_asm 2.16.f_bk $3$ (not in LMFDB) 2.16.ab_bg $4$ (not in LMFDB) 2.16.b_bg $4$ (not in LMFDB) 2.16.af_bk $6$ (not in LMFDB) 2.16.ad_bc $6$ (not in LMFDB) 2.16.d_bc $6$ (not in LMFDB)