addition on finite elliptic curves I tried to calculate the sum of two Points on an elliptic curve in a finite field. The Curve is defined as following:

$$y^2 \equiv x^3 + x \mod 257$$

So the curve parameters are $a = 1,b = 0,p = 257$.I want to add the two points $P = (1|60)$ and $Q = (15|7)$:

$$60^2 \equiv 1^3 + 1 \equiv 4 \mod 257$$
$$7^2 \equiv 15^3 + 15 \equiv 49 \mod 257$$

To add this two points, I calculate the slope $s$, $x_r$ and $y_r$:

$$s = \frac{y_q - y_p}{x_q - x_p}$$
$$x_r = s^2 - (x_p + x_q)$$
$$y_r = s(x_p - x_r) - y_p$$

for my points $P$ and $Q$ I get:

$$s = \frac{7-60}{15-1} = \frac{-53}{14} = -3.7857142857$$
$$x_r = (\frac{-53}{14})^2 - (1 + 15) = 14.3316326531 - 16 = -1.6683673469$$
$$y_r = \frac{-53}{14}(1-(-1.6683673469)) - 60 = -70.1016763848$$
$$R = (-1.6683673469|-70.1016763848)$$
However:
$$x_r^3 + x_r = -6.3121837787$$
while
$$y_r^2 = 4914.2450319648$$
So $R$ seems to be outside the curve. Even if I calculate everything modulo 257 I get weird results.

What I don't understand:


since I defined the curve on a finite field, I expected integer results
I expected $R$ to be part of the curve
the definition of $x_r$ is the difference between $s^2$ and the sum of the x-coordinates of the two added points. Why can $s^2$ be interpreted as an x-coordinate or even a length?
in reference to wikipedia, the addition of two points of an elliptic curve is commutative. That seems to be true for $x_r$, but why should it be true for $y_r$?


Notes:


I looked for an answer to this questions and all I did find were this and this. Both questions looked similar to mine but I still don't understand addition on elliptic curves.
I'm really new to elliptic curves, sorry if I did huge mistakes by calculating these values
You are right, when you're working on a finite field, you should get integer results. So let's look at why you aren't.


The problem lies in how you handle the fractions inside the equation, e.g.
$s=\frac{y_q-y_p}{x_q-x_p}$. Elliptic curve operations make use of finite field arithmetic. Maybe it'd be easier for you to grasp if you imagined the equation to look like $s=(y_q-y_p)(x_q-x_p)^{-1}\mod p$. This means that you have to calculate the modular inverse of the denominator of your fracture and then multiply this with the nominator.


$R$ is indeed part of the curve. Recalculate it using the "technique" described above.
It's not right to see it as length. As you're reducing the result modulo a prime ($\mod p$) the information which you call "length" is lost.
Take two points that are part of an elliptic curve, e.g. $P$ and $Q$. Commutative means that $P + Q$ gives the same result as $Q + P$. However it does not mean that you can just switch the $x$- and $y$-coordinates