A logarithmic identity linking \(π\) and the\\ \(j\)-invariant at a CM point

We present an exact logarithmic identity that connects \(π\), the naturallogarithm of the integer \(640320³+744\), and an infinite series oflogarithms whose terms involve the base \(-e-π√163\).The formula emerges from the evaluation of the elliptic modular invariant\(j\) at the CM point \(τ = {1+√-163}{2}\) and convergesat a spectacular speed, gaining about \(385\) decimal digits per term.This elementary yet striking relation illustrates how deep arithmeticinformation is encoded in the \(q\)-expansion of modular functions. {equation}{eq:main_abstract}{\,ln\!(640320³+744)\;=\;π√163\;+\;24∑ₙ₌₁∞ln\!({1}{1-(-e-π√163)ⁿ})\,}.{equation}