The Ramanujan-Serre differential operators and certain elliptic curves (Q2845418)

In this article, for each newform of weight \(2\) which is an eta-quotient listed in \textit{Y. Martin} and \textit{K. Ono} [Proc. Am. Math. Soc. 125, No.11, 3169--3176 (1997; Zbl 0894.11020)], the authors give a modular parametrization of the elliptic curve associated with it by using the Ramanujan-Serre differential operator. Let \(\Gamma_0(N)\) be the Hecke congruence subgroup of level \(N\). Put \(\Delta_N(\tau)=\left(\prod_{d\mid N}\eta(d\tau)\right)^{24/\mu_N}\), where \(\eta\) is the Dedekind eta-function and \(\mu_N\) is the index of \(\Gamma_0(N)\) in \(\text{SL}_2(\mathbb Z)\). Then \(\Delta_N\) is a cusp form of even integral weight \(k_N\) on \(\Gamma_0(N)\) if and only if \(N=1,2,3,5,6,11,14\) and \(15\). For these \(N\), \(\Delta_N(k_N\tau/2)^{2/k_N}\) are newforms of weight \(2\) of level \(k_N^2N/4\). The authors define the differential operator by NEWLINE\[NEWLINE\partial ^{(N)}_k(f)=(k_N/4)q\frac{df}{dq}-(k/4)f\cdot q\frac{d}{dq}\log \Delta_NNEWLINE\]NEWLINE on the modular forms \(f\) of weight \(k\) on \(\Gamma_0(N)\) with \(q=e^{2\pi i\tau}\). They found a modular form \(Q_N\) of weight \(4\) on \(\Gamma_0(N)\) such that the pair \(\left(Q_N/\Delta_N^{4/k_N},\partial^{(N)}_4(Q_N)/\Delta_N^{6/k_N}\right)\) is a modular parametrization of an elliptic curve associated with \(\Delta_N(k_N\tau/2)^{2/k_N}\). For \(N=1,2,3,5,6\), \(Q_N\) is any one of the Eisenstein series of weight \(4\) associated to cusps of \(\Gamma_0(N)\). Further for \(N=1,2,5,6\), they obtain similar results for the group conjugate to \(\Gamma_0(N)\) by the matrix \(\left(\begin{smallmatrix} 2&1\\0&2\end{smallmatrix}\right)\) by considering \(\Delta_N^\sharp(\tau)=-\Delta_N(\tau+1/2)\) instead of \(\Delta_N\). The newforms \(\Delta_N(k_N\tau/2)^{2/k_N}\) and \(\Delta_N^\sharp(k_N\tau/2)^{2/k_N}\) constitute the list of Martin and Ono (loc. cit.). For a related result, refer to Kazalicki, Sakai and Tasaka [\textit{M. Kazalicki} et al., Acta Arith. 163, No. 1, 33--43 (2014; Zbl 1300.11062)].