On page 104 of the paperback edition of Fearless Symmetry by Avner Ash and Robert Gross, they give one way of thinking of elliptic curves as $y^2=x^3+Ax+B$ wherein $A$ and $B$ can be any fixed integers so long as $2(4A^3+27B^2)\neq0$. Later, when discussing the example $y^2=x^3+1$, (so, an example wherein $A=0$ and $B=1$), they talk about how the above restrics the choices of number system $R$ we can make when looking for solution sets $E(R)$. It's probably best if I just quote directly (from the following p. 105):
"However, when we plug a number system $R$ into $E$ to get $E(R)$, we are tacitly assuming that $2(4A^3+27B^2)\neq0$ in $R$. So in our example we must assume that $6\neq0$ in $R$, so for instance $R=\mathbb F_2$ or $R=\mathbb F_3$ are not allowed for this $E$."
For the life of me, I can't imagine where the $6$ in that quote comes from. Obviously $54$—the actual calculation of the restriction—is a multiple of $6$, but I haven't seen any indications that that should matter, and I think there are situations in which the distinction might make a difference. For example—although I'll fully admit that I don't totally understand fields of the $\mathbb F_{p^n}$ format—I'm fairly certain that $6\neq0$ in $\mathbb F_{27}$, which, according to the quoted passage, would make $E(\mathbb F_{27})$ a valid answer set. However, I think that $54=0$ in $\mathbb F_{27}$, which would make $E(\mathbb F_{27})$ invalid.
What am I missing?