# What is the right basis of solutions of the Picard-Fuchs equation of the Legendre family around 0?

I have been trying to reconstruct some elliptic curves theory computationally and have gotten stuck on some period computations.

Specifically, let $$E_\lambda:\ y^2=x(x-1)(x-\lambda)$$ be the Legendre family of elliptic curves. The period integrals, (chosen to be) defined as $$\Omega=\Omega_\gamma(\lambda):=\int_\gamma \frac{dx}{\sqrt{x(x-1)(x-\lambda)}},$$ satisfy a second order differential equation, known as the Picard-Fuchs equation: $$4\lambda(1-\lambda)\frac{d^2\Omega}{d\lambda^2}+4(1-2\lambda)\frac{d\Omega}{d\lambda}-\Omega=0.$$

Question. Is there a basis, $$\{\Omega_1(\lambda), \Omega_2(\lambda)\}$$, of solutions around 0 for the Picard-Fuchs equation such that $$E_\lambda(\mathbb{C})\cong \mathbb{C}/(\mathbb{Z}+\frac{\Omega_2(\lambda)}{\Omega_1(\lambda)}\mathbb{Z})$$ for non-zero $$\lambda$$ (close to 0)?

Classic theory says that there is a unique holomorphic solution satisfying $$\Omega(0)=1$$, which is well known to be the hypergeometric function $$\Omega_1(\lambda)={}_2F_1(1/2, 1/2; 1; \lambda)=\sum_{n\ge0}\binom{-1/2}{n}^2\lambda^n.$$

Classic theory then tells us that there is a unique holomorphic function satisfying $$\rho(0)=0$$ and such that $$\Omega_2(\lambda)=C\left(\Omega_1(\lambda)\log{\lambda} + \rho(\lambda)\right)$$ is also a solution of the Picard-Fuchs equation, for all non-zero constants $$C$$, and that $$\Omega_1, \Omega_2$$ form a basis of solutions of the equation ($$\Omega_2$$ is a multi-valued map because of the appearance of $$\log$$). This $$\rho$$ is $$F_1(1/2,1/2;\lambda)$$, and is sometimes referred to as Fricke's hypergeometric function.

I have tried using these two solutions, $$\Omega_1$$ and $$\Omega_2$$, for different values of $$C$$, but I couldn't get the isomorphism in the question to be true. The most attractive value for $$C$$ is $$\frac{1}{2\pi i}$$, since then $$\mathbb{Z}+\frac{\Omega_2(\lambda)}{\Omega_1(\lambda)}\mathbb{Z} = \mathbb{Z}+\frac{1}{2\pi i}\left(\log{\lambda}+\frac{\rho(\lambda)}{\Omega_1(\lambda)}\right)$$ and we can take a single value for the $$\log$$ term.

But for $$\lambda=1/2$$, we have: $$\frac{1}{2\pi i}\left(\log{\lambda}+\frac{\rho(\lambda)}{\Omega_1(\lambda)}\right) =0.058728i$$ which corresponds to $$17.027i$$ in the fundamental domain for $$SL_2(\mathbb{Z})$$, which doesn't sit well with the fact that $$E_{1/2}\cong\mathbb{C}/(\mathbb{Z}+i\mathbb{Z}).$$