Thanks to technologies such as virtual network function the Fifth Generation (5G) of mobile networks dynamically allocate resources to different types of users in an on-demand fashion. Virtualization extends up to the 5G core, where software-defined networks and network slicing implement a customizable environment. These technologies can be controlled via application programming interfaces and web technologies, inheriting hence their security risks and settings. An attacker exploiting vulnerable implementations of the 5G core may gain privileged control of the network assets and disrupt its availability. However, there is currently no security assessment of the web security of the 5G core network. In this paper, we present the first security assessment of the 5G core from a web security perspective. We use the STRIDE threat modeling approach to define a complete list of possible threat vectors and associated attacks. Thanks to a suite of security testing tools, we cover all of these threats and test the security of the 5G core. In particular, we test the three most relevant open-source 5G core implementations, i.e., Open5GS, Free5Gc, and OpenAirInterface. Our analysis shows that all these cores are vulnerable to at least two of our identified attack vectors, demanding increased security measures in the development of future 5G core networks.