This paper, building on the work of Wiles and Taylor-Wiles, aims to prove two theorems regarding elliptic curves over the rational numbers. The first theorem states that every elliptic curve over \(\mathbf{Q}\) is modular. The second theorem asserts that if \(\overline{\rho}: \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_5)\) is an irreducible continuous representation with cyclotomic determinant, then \(\overline{\rho}\) is modular. The authors outline the methods used to prove these theorems, including the use of modular curves, deformation theory, and local computations. They also discuss the reduction of types for admissible representations and the construction of universal deformations. The paper concludes with detailed proofs for specific cases, particularly focusing on the wild 3-adic exercises.This paper, building on the work of Wiles and Taylor-Wiles, aims to prove two theorems regarding elliptic curves over the rational numbers. The first theorem states that every elliptic curve over \(\mathbf{Q}\) is modular. The second theorem asserts that if \(\overline{\rho}: \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_5)\) is an irreducible continuous representation with cyclotomic determinant, then \(\overline{\rho}\) is modular. The authors outline the methods used to prove these theorems, including the use of modular curves, deformation theory, and local computations. They also discuss the reduction of types for admissible representations and the construction of universal deformations. The paper concludes with detailed proofs for specific cases, particularly focusing on the wild 3-adic exercises.