Modulation classification (MC) is the first step performed at the receiver side unless the modulation type is explicitly indicated by the transmitter. Machine learning techniques have been widely used for MC recently. In this paper, we propose a novel MC technique dubbed as Joint Equalization and Modulation Classification based on Constellation Network (EMC²-Net). Unlike prior works that considered the constellation points as an image, the proposed EMC²-Net directly uses a set of 2D constellation points to perform MC. In order to obtain clear and concrete constellation despite multipath fading channels, the proposed EMC²-Net consists of equalizer and classifier having separate and explainable roles via novel three-phase training and noise-curriculum pretraining. Numerical results with linear modulation types under different channel models show that the proposed EMC²-Net achieves the performance of state-of-the-art MC techniques with significantly less complexity.