This differential equation has elliptic, or parabolic or hyperbolic solutions. Leaves lanceolate to elliptic or ovate, usually about . The Pythagorean theorem fails in elliptic geometry. A paraboloid is either elliptic or hyperbolic. Spherical geometry is the simplest form of elliptic geometry. Historically, elliptic functions were discovered as inverse functions of elliptic integrals. These cases give rise to the hyperbolic, parabolic or elliptic cylinders respectively. That hand wrote a number of elliptic integrals. They are simple, membranous, orbicular to elliptic and acuminate. This would conflict with the modularity theorem, which asserted that all elliptic curves are modular. The security of elliptic curve cryptography is based on number theoretic problems involving elliptic curves. There are still other conventions for the notation of elliptic integrals employed in the literature. Point at infinity is the identity element of elliptic curve arithmetic. The conjecture relies on analytic and arithmetic objects defined by the elliptic curve in question. He was also an innovator in the field of elliptic functions, discoverer of Abelian functions.