Elliptic Curves: Construction and Properties

• Creation of an elliptic curve over a field
• Creation of a curve with given j-invariant
• Models: Weierstrass form, integral model, minimal model, simplified model
• Arithmetic with rational points
• Extension and lifting of curves induced by maps of base rings
• Function field of the curve
• Invariants: b-invariants, c-invariants, j-invariant, discriminant
• Division polynomials, m-torsion subgroups
• Subgroups and subschemes of elliptic curves as separate types