Elliptic Curves: Operations over $\mbox{\bf Q}$

• Standard models and conversions
• Construction from plane curves
• Invariants: conductor, regulator, periods, and period lattice
• Tamagawa numbers
• Tate's algorithm for computing local information, Kodaira symbols
• Twist of an elliptic curve by an integer
• Heights: local height, naive height, canonical height, height pairing
• Elliptic logarithm and p-adic elliptic logarithm
• Mordell-Weil rank, bounds on Mordell-Weil rank, analytic rank
• Mordell-Weil group and torsion subgroup
• S-integral points: determination of the finite set of affine points with denominator generated over the finite set of primes S.
• John Cremona's database of all elliptic curves over Q having conductor up to $10\,000$

Standard models are provided together with heights and invariants. For curves over Q invariants such as conductor, regulator, and local information (Tate's algorithm) are available. The Mordell-Weil rank and group is computed using code based on John Cremona's SPMquotMWRANK" program. Magma takes 140 seconds to determine that the curve

y² + xy = x³ - 215x + 1192

has group $\mbox{\bf Z}_2\oplus\mbox{\bf Z}\oplus\mbox{\bf Z}$, and 100 seconds to determine that the curve

$$y^2 = x^3 + 36\,861\,504\,658\,225x^2 + 1\,807\,580\,157\,674\,409\,809\,510\,400x$$

has rank 13.