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Constructing elliptic curves

  1. Feb 10, 2013 #1
    I'm trying to build intuition on how to construct elliptic curves satisfying some predetermined properties. I'm working on these two cases atm:

    1) I can construct an elliptic curve with a point of order n, but how should I proceed if I want to ensure that the curve has also rank > 0? (I.e. I want also a point of inf order). Is there a standard method to do so?
    2) In computing the order of the tors group, I'm interested in those curves for which knowing that #E(Q)|#E(F_p) is not sufficient to determine #E(Q). Does anyone have an example of such a curve? In general, how can one construct elliptic curves with this property?

    Any help with either 1 or 2 is much appreciated
  2. jcsd
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