when E is an elliptic curve, $E:F(x,y)=y^2+a_1x y+a_3y-x^3-a_2x^2-a_4x-a_6=0$
and O=[0,1,0] is the base point, then $ord_O(x)=-2,ord_O(y)=-3$.
why is it? please teach me how to calculate the order.
it is written on p48.proposition1.5 of the book(GTM106,the arithmetic of elliptic curves).
This is the order of a function at a point on an elliptic curve. We really should use projective coordinates, so the equation is $F(X,Y,Z)=Y^2Z+a_1XYZ+\cdots=0$. Then $x=X/Z$ is a rational function on the projective plane with divisor $(L_1)-(L_2)$ where $L_1$ is the line $X=0$ and $L_2$ is the line $Z=0$. If $P$ is a point on $E$ then $\textrm{ord}_P(x)=i_P(E,L_1)-i_P(E,L_2)$ where $i_P(E,L)$ is the intersection multiplicity of $E$ and $L$ at the point $P$. Here $i_O(E,L_2)=3$ as the point $O$ is an inflection point, and $L_2$ (the line at infinity) is the tangent to $E$ at $O$. Also $L_1$ passes through $O$ but is not a tangent, so $i_p(E,L_1)=1$.
The calculation for $\textrm{ord}_O(y)$ is similar, but a little bit easier.