Low-rank adaptation (LoRA) is a widely used parameter-efficient fine-tuning method that places trainable low-rank adapters into frozen pre-trained models. Recent studies show that using fewer LoRA adapters may still maintain or even improve performance, but existing methods still distribute adapters broadly, leaving \emph{where to place a limited number of adapters to maximize performance} largely open. To investigate this, we introduce \textbf{PAGE} (\textbf{P}rojected \textbf{A}dapter \textbf{G}radient \textbf{E}nergy), a gradient-based sensitivity probe that estimates the initial trainable gradient energy available to each candidate LoRA adapter. Surprisingly, we find that PAGE is highly concentrated on a single shallow FFN down-projection across two model families and four downstream tasks. We term this module the \textbf{dominant adaptation module} and show that its layer index is architecture-dependent but task-stable. Motivated by this finding, we propose \textbf{DomLoRA}, a placement method that places a single adapter at the dominant adaptation module. With only \textbf{0.7\%} of vanilla LoRA's trainable parameters, DomLoRA outperforms it on average across downstream tasks, including instruction following, mathematical reasoning, coding, and multi-turn conversation. This method also matches or improves other LoRA variants and reduces training time by up to \textbf{2.74}$\times$ compared with broad placement, supporting the dominant adaptation module perspective as a practical placement guideline.

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