Math 502
Maple Routines for Cubic Curves

due Tuesday, April 15, 2008

The cubic curve routines for Maple may be found at

winitell

 

For these routines a cubic curve is represented by its vector of ten coefficients. For reasons connected with the usual numbering of the coefficients of a cubic curve in Weierstrass normal form the vector of coefficients is stored as an array with index values in the interval $\left[-3,6\right]$. Of course such an array may be introduced explicitly using the Maple function array. The function winitell is provided here for convenience in entering a cubic curve in Weierstrass form. It takes as argument a list with 6 entries which are the coefficients $a_0,a_1,a_2,a_3,a_4,a_6$. (The automatic value $a_5=1$ serves behind the scenes in the 10 slot array as the coefficient of $y^2$.) There is an optional second argument, a prime number $p$, that is used to indicate the field of coefficients, for handling arithmetic mod $p$.

affelleq

 
If $E$ is (the coefficient array for) a cubic curve in Weierstrass form, then affelleq(E) returns its affine equation relative to coordinates $X,Y$. (These coordinate symbols are used globally by this collection of routines.)

Example:

 E:=winitell([1,0,0,0,-25,0]): affelleq(E);

                                 2    3

                                Y  = X  - 25 X

isoncurve

 
If $E$ is a cubic curve and $P$ a point given either in affine form (a pair) or homogeneous form (a triple), the function isoncurve may be called with $E$ as first argument and $P$ as second argument to determine “true” or “false” for the question of whether $P$ lies on $E$. An optional third argument $p$, a prime, indicates the question should be considered for arithmetic mod $p$.

Example:

F:=winitell([1,0,0,0,5,19]): affelleq(F);

                               2    3

                              Y  = X  + 5 X + 19

 

isoncurve(F,[-2,1]); isoncurve(F,[2,-1]);

                              true

                              false

negell

 
If $E$ is a cubic curve and $P$ a point on $E$, then the function negell may be called with $E$ as first argument and $P$ as second argument (and with a prime $p$ as optional third argument) to find the negative of $P$ relative to the arithmetic on $E$. Note that this type of negative of a point is different from the negative of a point in vector arithmetic.

Example:

negell(F,[-2,1]);

                              [-2,-1]

negell(F,[2,-1]);

                         negell: Point is not on curve

powell

 
The function powell may be used to find an integer multiple of a point relative to the arithmetic on $E$, i.e., $$nP=\underset{n\text{times}}{P+\dots +P}\text{.}$$ Note that this type of scalar multiple is different from the scalar multiple of a point in vector arithmetic. Call powell with the curve as first argument, the integer multiplier as second argument, and the point as third argument. (An optional fourth argument $p$, a prime, indicates computation mod $p$.)

Example:

powell(F, 3, [-2,1]);

                               -153306  83099195

                              [-------, --------]

                                97969   30664297

Note the entry of denominators. It is the rule rather than the exception when working with rational coefficients and coordinates. Of course, denominators should not appear in computations modulo a prime $p$.

Example:

powell(F, 3, [-2,1], 7);

                              [2, 3]

 

isoncurve(F, [2,3]); isoncurve(F, [2,3], 7);

                              false

                              true

addell

 
Use addell to add points of a cubic curve $E$ relative to the arithmetic of points on the curve. Call addell with the curve as first argument, the points to be added as second and third arguments, and a prime $p$, if wanted, as optional fourth argument.

Example:

 isoncurve(F,[5,13]);

                              true

 

addell(F,[-2,1],[5,13]);

                                   -3  -1483

                                  [--, -----]

                                   49   343

Get used to seeing denominators. Note also that is it not an easy matter to come up with examples of points on $E$ having rational coordinates, much less integer coordinates.

But now explain what is happening here:

addell(F,[-2,1],[5,13],7);

                                  [0, 1, 0]

There are 3 coordinates, because the point of $F$ in question is not a point of the affine plane. It is, in fact, the triple of homogeneous coordinates for the unique point of $F$ on the line at infinity — the origin for the arithmetic on $F$. This computation means, therefore, that $\left(5,13\right)$ is the negative relative to the arithmetic on $F$ when coefficients and coordinates are integers mod $7$. Note that this is consistent with the fact that $\left(-2,-1\right)$ is the negative of $\left(-2,1\right)$ when coefficients and coordinates are rational since one has $$5\equiv -2\left(mod7\right)\text{and}13\equiv -1\left(mod7\right)\text{.}$$