# Properties

 Label 2.256.ach_cay Base Field $\F_{2^{8}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2^{8}}$ Dimension: $2$ L-polynomial: $( 1 - 16 x )^{2}( 1 - 27 x + 256 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.180343027596$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 51750 4247433000 281355229086750 18446541825576618000 1208925710564174413593750 79228162202161824397479843000 5192296846117850074003925016156750 340282366790830915872938048362603188000 22300745197636817990040624128736126131751750 1461501637326077011928629537952148728325508125000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 198 64808 16770078 4294920208 1099511528598 281474975601848 72057593865607758 18446744066656407328 4722366482680374610758 1208925819610637278597448

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
 The isogeny class factors as 1.256.abg $\times$ 1.256.abb and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.256.abg : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.256.abb : $$\Q(\sqrt{-295})$$.
All geometric endomorphisms are defined over $\F_{2^{8}}$.

## 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 2.256.af_ano $2$ (not in LMFDB) 2.256.f_ano $2$ (not in LMFDB) 2.256.ch_cay $2$ (not in LMFDB) 2.256.al_dc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.256.af_ano $2$ (not in LMFDB) 2.256.f_ano $2$ (not in LMFDB) 2.256.ch_cay $2$ (not in LMFDB) 2.256.al_dc $3$ (not in LMFDB) 2.256.abb_ts $4$ (not in LMFDB) 2.256.bb_ts $4$ (not in LMFDB) 2.256.abr_bki $6$ (not in LMFDB) 2.256.l_dc $6$ (not in LMFDB) 2.256.br_bki $6$ (not in LMFDB)