Elliptic Curves over $\mathbb{Q}$

• COMPUTATION:
  • Play around with each of the above groups in SAGE . Here are examples of how to compute with each of the groups listed above in SAGE to get you started. Be creative and experiment!

    sage: ZZ
    Integer Ring
    sage: a = ZZ(5); b = ZZ(7); a+b
    12
    
    sage: QQ
    Rational Field
    sage: QQ(3/4) + QQ(2/3)
    17/12
    
    sage: R = IntegerModRing(12)
    sage: print R
    Ring of integers modulo 12
    sage: R(7) + R(8)
    3
    
    sage: R = FiniteField(7)
    sage: print R
    Finite Field of size 7
    sage: R(4)*R(5)
    6
    

  • Compute with elliptic curves over $\mathbb{Q}$ . Here is an example to get you started.

    sage: E = EllipticCurve([-36,0])
    sage: E
    Elliptic Curve defined by y^2  = x^3 - 36*x over Rational Field
    sage: P = E([-3,9])   # or use E.gens(proof=False)
    sage: P + P
    (25/4 : -35/8 : 1)
    

  • Draw some graphs of elliptic curves (using the new program I just wrote!). Here is an example to get you started:

    sage: E = EllipticCurve([-36,0])
    sage: P = plot(E,rgbcolor=(0,0,1))
    sage: pnt = E([-3,9])
    sage: pnt2 = 2*pnt
    sage: c1 = point(pnt, pointsize=100)
    sage: c2 = point(pnt2, rgbcolor=(1,0,0), pointsize=100)
    sage: show(P + c1 + c2)
    

• THEORY:
  • Let $E$ be an elliptic curve given by an equation $y^2=x^3+ax+b$ . Prove that the following steps can be used to compute $+$ on $E$ : Given $P_1, P_2$ , this algorithm computes the third point $R=P_1+P_2$ .
    1. [Is $P_i=\O$ ?] If $P_1=\O$ set $R=P_2$ or if $P_2=\O$ set $R=P_1$ and terminate. Otherwise write $(x_i,y_i)=P_i$ .
    2. [Negatives] If $x_1 = x_2$ and $y_1 = -y_2$ , set $R=\O$ and terminate.
    3. [Compute $\lambda$ ]Set $$\lambda = \begin{cases} (3x_1^2+a)/(2y_1) & \text{if }P_1 = P_2,\\ (y_1-y_2)/(x_1-x_2) & \text{otherwise.} \end{cases}$$
       
    4. [Compute Sum] Then $R = \displaystyle \left(\lambda^2 -x_1 - x_2, -\lambda x_3 - \nu\right)$ , where $\nu = y_1 - \lambda x_1$ and $x_3=\lambda^2 -x_1 - x_2$ is the $x$ -coordinate of $R$ .
• RESEARCH:
  • Using the Internet find two web pages that define abelian groups. Understand the definitions (perhaps by copying them down onto a piece of paper). Do they agree with the definition that I gave above?