Algebraic Geometry

Point counting on curves

How do you count points on an elliptic curve over a finite field in Sage?

Over prime finite fields, includes both the baby step giant step method and the SEA (Schoof-Elkies-Atkin) algorithm (implemented in PARI by Christophe Doche and Sylvain Duquesne). An example taken form the Reference manual:

sage: E = EllipticCurve(GF(10007),[1,2,3,4,5])
sage: E.cardinality()
10076

The command E.points() will return the actual list of rational points.

How do you count points on a plane curve over a finite field? The rational_points command produces points by a simple enumeration algorithm. Here is an example of the syntax:

sage: x,y,z = PolynomialRing(GF(5), 3, 'xyz').gens()
sage: C = Curve(y^2*z^7 - x^9 - x*z^8); C
Projective Plane Curve over Finite Field of size 5 defined by -x^9 + y^2*z^7 - x*z^8
sage: C.rational_points()
[(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1), (3 : 1 : 1), (3 : 4 : 1)]
sage: C.rational_points(algorithm="bn")
[(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1), (3 : 1 : 1), (3 : 4 : 1)]

The option algorithm="bn uses Sage’s Singular interface and calls the brnoeth package.

Here is another example using Sage’s rational_points applied to Klein’s quartic over \(GF(8)\).

sage: x, y, z = PolynomialRing(GF(8,'a'), 3, 'xyz').gens()
sage: f = x^3*y+y^3*z+x*z^3
sage: C = Curve(f); C
Projective Plane Curve over Finite Field in a of size 2^3 defined by x^3*y + y^3*z + x*z^3
sage: C.rational_points()
[(0 : 0 : 1),
 (0 : 1 : 0),
 (1 : 0 : 0),
 (1 : a : 1),
 (1 : a^2 : 1),
 (1 : a^2 + a : 1),
 (a : 1 : 1),
 (a : a^2 : 1),
 (a : a^2 + 1 : 1),
 (a + 1 : a + 1 : 1),
 (a + 1 : a^2 : 1),
 (a + 1 : a^2 + a + 1 : 1),
 (a^2 : 1 : 1),
 (a^2 : a^2 + a : 1),
 (a^2 : a^2 + a + 1 : 1),
 (a^2 + 1 : a + 1 : 1),
 (a^2 + 1 : a^2 + 1 : 1),
 (a^2 + 1 : a^2 + a : 1),
 (a^2 + a : 1 : 1),
 (a^2 + a : a : 1),
 (a^2 + a : a + 1 : 1),
 (a^2 + a + 1 : a : 1),
 (a^2 + a + 1 : a^2 + 1 : 1),
 (a^2 + a + 1 : a^2 + a + 1 : 1)]

Other methods

• For a plane curve, you can use Singular’s closed_points command. The input is the vanishing ideal \(I\) of the curve \(X\) in a ring of \(2\) variables \(F[x,y]\). The closed_points command returns a list of prime ideals (each a Gröbner basis), corresponding to the (distinct affine closed) points of \(V(I)\). Here’s an example:

  sage: singular_console()
                       SINGULAR                             /  Development
   A Computer Algebra System for Polynomial Computations   /   version 3-0-1
                                                         0<
       by: G.-M. Greuel, G. Pfister, H. Schoenemann        \   October 2005
  FB Mathematik der Universitaet, D-67653 Kaiserslautern    \
  // ** executing /home/wdj/sagefiles/sage-0.9.4/local/LIB/.singularrc
  > LIB "brnoeth.lib";
  > ring s = 2,(x,y),lp;
  > ideal I = x4+x,y4+y;
  > list L = closed_points(I);
  > L;
  [1]:
     _[1] = y
     _[2] = x
  [2]:
     _[1] = y
     _[2] = x+1
  [3]:
     _[1] = y
     _[2] = x2+x+1
  [4]:
     _[1] = y+1
     _[2] = x
  [5]:
     _[1] = y+1
     _[2] = x+1
  [6]:
     _[1] = y+1
     _[2] = x2+x+1
  [7]:
     _[1] = y2+y+1
     _[2] = x+1
  [8]:
     _[1] = y2+y+1
     _[2] = x
  [9]:
     _[1] = y2+y+1
     _[2] = x+y
  [10]:
     _[1] = y2+y+1
     _[2] = x+y+1
  > Auf Wiedersehen.
  

  sage: singular.lib("brnoeth.lib")
  sage: s = singular.ring(2,'(x,y)','lp')
  sage: I = singular.ideal('x^4+x', 'y^4+y')
  sage: L = singular.closed_points(I)
  sage: # Here you have all the points :
  sage: L       # random
  [1]:
     _[1]=y+1
     _[2]=x+1
  ...
  sage: l=[L[k].sage() for k in [1..10]]; len(l) # there are 10 points
  10
  sage: r=sorted(l[0].ring().gens()); r
  [y, x]
  sage: r in [t.gens() for t in l] #  one of them is given by [y,x]
  True
  

• Another way to compute rational points is to use Singular’s NSplaces command. Here’s the Klein quartic over \(GF(8)\) done this way:

  sage: singular.LIB("brnoeth.lib")
  sage: s = singular.ring(2,'(x,y)','lp')
  ...
  sage: f = singular.poly('x3y+y3+x')
  ...
  sage: klein1 = f.Adj_div(); print(klein1)
  [1]:
     [1]:
        //   coefficients: ZZ/2
  //   number of vars : 2
  //        block   1 : ordering lp
  //                  : names    x y
  //        block   2 : ordering C
  ...
  sage: # define a curve X = {f = 0} over GF(2)
  sage: klein2 = singular.NSplaces(3,klein1)
  sage: print(singular.eval('extcurve(3,%s)'%klein2.name()))
  Total number of rational places : NrRatPl = 23
  ...
  sage: klein3 = singular.extcurve(3, klein2)
  

  Above we defined a curve \(X = \{f = 0\}\) over \(GF(8)\) in Singular.

  sage: print(klein1)
  [1]:
     [1]:
        //   coefficients: ZZ/2
  //   number of vars : 2
  //        block   1 : ordering lp
  //                  : names    x y
  //        block   2 : ordering C
     [2]:
        //   coefficients: ZZ/2
  //   number of vars : 3
  //        block   1 : ordering lp
  //                  : names    x y z
  //        block   2 : ordering C
  [2]:
     4,3
  [3]:
     [1]:
        1,1
     [2]:
        1,2
  [4]:
     0
  [5]:
     [1]:
        [1]:
           //   coefficients: ZZ/2
  //   number of vars : 3
  //        block   1 : ordering ls
  //                  : names    x y t
  //        block   2 : ordering C
        [2]:
           1,1
  sage: print(klein1[3])
  [1]:
     1,1
  [2]:
     1,2
  

  For the places of degree \(3\):

  sage: print(klein2[3])
  [1]:
     1,1
  [2]:
     1,2
  [3]:
     3,1
  [4]:
     3,2
  [5]:
     3,3
  [6]:
     3,4
  [7]:
     3,5
  [8]:
     3,6
  [9]:
     3,7
  

  Each point below is a pair: (degree, point index number).

  sage: print(klein3[3])
  [1]:
     1,1
  [2]:
     1,2
  [3]:
     3,1
  [4]:
     3,2
  [5]:
     3,3
  [6]:
     3,4
  [7]:
     3,5
  [8]:
     3,6
  [9]:
     3,7
  

  To actually get the points of \(X(GF(8))\):

  sage: R = klein3[1][5]
  sage: R.set_ring()
  sage: singular("POINTS;")
  [1]:
     [1]:
        0
     [2]:
        1
     [3]:
        0
  [2]:
     [1]:
        1
     [2]:
        0
     [3]:
        0
  ...
  

  plus 21 others (omitted). There are a total of \(23\) rational points.

Riemann-Roch spaces using Singular

To compute a basis of the Riemann-Roch space of a divisor \(D\) on a curve over a field \(F\), one can use Sage’s wrapper riemann_roch_basis of Singular’s implementation of the Brill Noether algorithm. Note that this wrapper currently only works when \(F\) is prime and the divisor \(D\) is supported on rational points. Below are examples of how to use riemann_roch_basis and how to use Singular itself to help an understanding of how the wrapper works.

• Using riemann_roch_basis:

  sage: x, y, z = PolynomialRing(GF(5), 3, 'xyz').gens()
  sage: f = x^7 + y^7 + z^7
  sage: X = Curve(f); pts = X.rational_points()
  sage: D = X.divisor([ (3, pts[0]), (-1,pts[1]), (10, pts[5]) ])
  sage: X.riemann_roch_basis(D)
  [(-2*x + y)/(x + y), (-x + z)/(x + y)]
  

• Using Singular’s BrillNoether command (for details see the section Brill-Noether in the Singular online documentation (http://www.singular.uni-kl.de/Manual/html/sing_960.htm and the paper {CF}):

  sage: singular.LIB('brnoeth.lib')
  sage: _ = singular.ring(5,'(x,y)','lp')
  sage: print(singular.eval("list X = Adj_div(-x5+y2+x);"))
  Computing affine singular points ...
  Computing all points at infinity ...
  Computing affine singular places ...
  Computing singular places at infinity ...
  Computing non-singular places at infinity ...
  Adjunction divisor computed successfully
  
  The genus of the curve is 2
  sage: print(singular.eval("X = NSplaces(1,X);"))
  Computing non-singular affine places of degree 1 ...
  sage: print(singular("X[3];"))
  [1]:
     1,1
  [2]:
     1,2
  [3]:
     1,3
  [4]:
     1,4
  [5]:
     1,5
  [6]:
     1,6
  

  The first integer of each pair in the above list is the degree \(d\) of a point. The second integer is the index of this point in the list POINTS of the ring X[5][\(d\)][1]. Note that the order of this latter list is different every time the algorithm is run, e.g. \(1\), \(1\) in the above list refers to a different rational point each time. A divisor is given by defining a list \(G\) of integers of the same length as X[3] such that if the \(k\)-th entry of X[3] is \(d\), \(i\), then the \(k\)-th entry of \(G\) is the multiplicity of the divisor at the \(i\)-th point in the list POINTS of the ring X[5][\(d\)][1]. Let us proceed by defining a “random” divisor of degree 12 and computing a basis of its Riemann-Roch space:

  sage: singular.eval("intvec G = 4,4,4,0,0,0;")
  ''
  sage: singular.eval("def R = X[1][2];")
  ''
  sage: singular.eval("setring R;")
  ''
  sage: print(singular.eval("list LG = BrillNoether(G,X);"))
  Forms of degree 6 :
  28
  
  Vector basis successfully computed
  

AG codes

Sage can compute an AG code \(C=C_X(D,E)\) by calling Singular’s BrillNoether to compute a basis of the Riemann Roch space \(L(D)=L_X(D)\). In addition to the curve \(X\) and the divisor \(D\), you must also specify the evaluation divisor \(E\).

Note that this section has not been updated since the wrapper riemann_roch_basis has been fixed. See above for how to properly define a divisor for Singular’s BrillNoether command.

Here’s an example, one which computes a generator matrix of an associated AG code. This time we use Singular’s AGCode_L command.

sage: singular.LIB('brnoeth.lib')
sage: singular.eval("ring s = 2,(x,y),lp;")
''
sage: print(singular.eval("list HC = Adj_div(x3+y2+y);"))
Computing affine singular points ...
Computing all points at infinity ...
Computing affine singular places ...
Computing singular places at infinity ...
Computing non-singular places at infinity ...
Adjunction divisor computed successfully

The genus of the curve is 1
sage: print(singular.eval("list HC1 = NSplaces(1..2,HC);"))
Computing non-singular affine places of degree 1 ...
Computing non-singular affine places of degree 2 ...
sage: print(singular.eval("HC = extcurve(2,HC1);"))
Total number of rational places : NrRatPl = 9

We set the following to junk to discard the output:

sage: junk = singular.eval("intvec G = 5;")      # the rational divisor G = 5*HC[3][1]
sage: junk = singular.eval("def R = HC[1][2];")
sage: singular.eval("setring R;")
''

The vector \(G\) represents the divisor “5 times the point at infinity”.

Next, we compute the Riemann-Roch space.

sage: print(singular.eval("BrillNoether(G,HC);"))
Forms of degree 3 :
10

Vector basis successfully computed

[1]:
   _[1]=x
   _[2]=z
[2]:
   _[1]=y
   _[2]=z
[3]:
   _[1]=1
   _[2]=1
[4]:
   _[1]=y2+yz
   _[2]=xz
[5]:
   _[1]=y3+y2z
   _[2]=x2z

That was the basis of the Riemann-Roch space, where each pair of functions represents the quotient (first function divided by second function). Each of these basis elements get evaluated at certain points to construct the generator matrix of the code. We next construct the points.

sage: singular.eval("def R = HC[1][5];")
'// ** redefining R **'
sage: singular.eval("setring R;")
''
sage: print(singular.eval("POINTS;"))
[1]:
   [1]:
      0
   [2]:
      1
   [3]:
      0
[2]:
   [1]:
      0
   [2]:
      1
   [3]:
      1
[3]:
   [1]:
      0
   [2]:
      0
   [3]:
      1
[4]:
   [1]:
      (a+1)
   [2]:
      (a)
   [3]:
      1
...

plus \(5\) more, for a total of \(9\) rational points on the curve. We define our “evaluation divisor” \(D\) using a subset of these points (all but the first):

sage: singular.eval("def ER = HC[1][4];")
''
sage: singular.eval("setring ER;")
''
sage: # D = sum of the rational places no. 2..9 over F_4
sage: singular.eval("intvec D = 2..9;")
''
sage: # let us construct the corresponding evaluation AG code :
sage: print(singular.eval("matrix C = AGcode_L(G,D,HC);"))
Forms of degree 3 :
10

Vector basis successfully computed

sage: # here is a linear code of type [8,5,> = 3] over F_4
sage: print(singular.eval("print(C);"))
0,0,(a+1),(a),  1,  1,    (a),  (a+1),
1,0,(a),  (a+1),(a),(a+1),(a),  (a+1),
1,1,1,    1,    1,  1,    1,    1,
0,0,(a),  (a+1),1,  1,    (a+1),(a),
0,0,1,    1,    (a),(a+1),(a+1),(a)

This is, finally, our desired generator matrix, where a represents a generator of the field extension of degree \(2\) over the base field \(GF(2)\).

Can this be “wrapped”?