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Re: Plot of Elliptic Curve with Grid

• To: mathgroup at smc.vnet.net
• Subject: [mg51537] Re: [mg51512] Plot of Elliptic Curve with Grid
• From: "David Park" <djmp at earthlink.net>
• Date: Thu, 21 Oct 2004 22:21:02 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

I made your plot using DrawGraphics as follows.

Needs["DrawGraphics`DrawingMaster`"]

I don't think your integer points corresponded to the curve you specified.
So I tried to find some and came up with the following list.

integerpoints = {{0, 1}, {0, -1}, {1, 2}, {1, -2}, {8, 23}, {8, -23}};
xpoints = Union@(First /@ integerpoints);
ypoints = Union@(Last /@ integerpoints);

I then made the plot with the following statement. The grid lines and tick
marks and labels match the values for the integer points.

Draw2D[
    {ImplicitDraw[y^2 == x^3 + 2x + 1, {x, -1, 9}],
      CirclePoint[#, 3, Black, Yellow] & /@ integerpoints},
    AspectRatio -> 1.5,
    Frame -> True,
    FrameTicks ->
       {CustomTicks[Identity, databased[xpoints]],
        CustomTicks[Identity, databased[ypoints]],
        CustomTicks[Identity, databased[xpoints], CTNumberFunction -> (""
&)],
        CustomTicks[Identity, databased[ypoints],
          CTNumberFunction -> ("" &)]},
    GridLines ->
       {CustomGridLines[Identity, databased[xpoints]],
        CustomGridLines[Identity, databased[ypoints]]},
    PlotLabel -> SequenceForm["Elliptic Curve ", y^2 == x^3 + 2x + 1],
    Background -> Linen,
    ImageSize -> 450];

I was going to send you privately the notebook and a gif image of the plot,
but taking a quick look at your email address I have no idea how it is
supposed to be decrypted. If you want to have a usable email account without
being bothered by spam or virus email subscribe to SpamArrest or some
similar service. It works.

David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/



From: flip [mailto:flip_alpha at safebunch.com]
To: mathgroup at smc.vnet.net

Hello,

I would like to plot an elliptic curve over Fp of the form:

y^2 = x^3 + ax + b (1)

I would then like to plot the list of points that satisfy (1).  {Note: I
have a way to generate that list).

I would like the continuous plot (like using implicit plot over reals) of
(1) with a grid having points of intersection over Fp (the integer points)
shown on the plot (over a grid).

Example:

y^2 = x^3 + 2x + 1 over F5

This curve has 7 points (counting the point at infinity).

The list of points is: S = {{0,1},{0,4},{1,2},{1,3},{3,2},{3,3}}

I would like to show a grid plot with the elliptic curve (continuous over
reals) superimposed over the discrete points given above (with points of
intersection (a dot of some sort shown for each point above)).

I would like to be able to pass in the "a, b, S" and have this automatically
generate the plot.

Is this easy?

Thanks for any input, Flip

****email****         flip         %%%%     @    %%%%%
nethere......com****************

Sorry for the crypto in my email, but spam is a killer