The ascending pharyngeal artery, the smallest branch of the external carotid, is a long, slender vessel, deeply seated in the neck, beneath the other branches of the external carotid and under the Stylopharyngeus. It lies just superior to the bifurcation of the common carotid artery

Common carotid artery

In human anatomy, the common carotid artery is an artery that supplies the head and neck with oxygenated blood; it divides in the neck to form the external and internal carotid arteries. - Structure :...

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Its branches are listed here: Pharyngeal branches of ascending pharyngeal artery

Pharyngeal branches of ascending pharyngeal artery

The pharyngeal branches of ascending pharyngeal artery are small branches of the ascending pharyngeal artery.They are three or four in number. Two of these descend to supply the Constrictores pharyngis medius and inferior and the Stylopharyngeus, ramifying in their substance and in the mucous...

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Course

It arises from the back part of the external carotid, near the commencement of that vessel, and ascends vertically between the internal carotid and the side of the pharynx

Pharynx

The human pharynx is the part of the throat situated immediately posterior to the mouth and nasal cavity, and anterior to the esophagus and larynx. The human pharynx is conventionally divided into three sections: the nasopharynx , the oropharynx , and the laryngopharynx...

, to the under surface of the base of the skull, lying on the Longus capitis.