Marcus Ulpius Traianus was born on 18 September 53AD in the Roman province of Hispania Baetica (in what is now Andalusia in modern Spain), in the city of Italica (now in the municipal area of Santiponce, in the outskirts of Seville). Although frequently designated the first provincial emperor, and dismissed by later writers such as Cassius Dio (himself of provincial origin) as "an Iberian, and neither an Italian nor even an Italiot", Trajan appears to have hailed on his father's side from the area of Tuder (modern Todi) in Umbria, at the border with Etruria, and on his mother's side from the Gens Marcia, of an Italic family of Sabine origin. Trajan's birthplace of Italica was founded as a Roman military colony of "Italian" settlers in 206BC, though it is unknown when the Ulpii arrived there. It is possible, but cannot be substantiated, that Trajan's ancestors married local women and lost their citizenship at some point, but they certainly recovered their status when the city became a municipium with Latin citizenship in the mid-1st century BC.
Under Charlemagne the Franks reached the height of their power. After campaigns against Lombards, Avars, Saxons, and Basques, the resulting Carolingian Empire stretched from the Pyrenees to Central Germany, from the North Sea to the Adriatic. In 800 the Pope made Charlemagne Emperor of the West in return for protection of the Church. The Carolingian Empire was a conscious effort to recreate a central administration modeled on that of the Roman Empire,[13] but the motivations behind military expansion differed. Charlemagne hoped to provide his nobles an incentive to fight by encouraging looting on campaign. Plunder and spoils of war were stronger temptations than imperial expansion, and several regions were invaded over and over in order to bolster the coffers of Frankish nobility.[14] Cavalry dominated the battlefields, and while the high costs associated with equipping horses and horse-riders helped limit their numbers, Carolingian armies maintained an average size of 20,000 by recruiting infantry from imperial territories near theaters of operation.[15] The Empire lasted from 800 to 843, when, following Frankish tradition, it was split between the sons of Louis the Pious by the Treaty of Verdun.
The imperial period of Rome lasted approximately 1,500 years compared to the 500 years of the Republican era. The first two centuries of the empire's existence were a period of unprecedented political stability and prosperity known as the Pax Romana, or ``Roman Peace ''. Following Octavian's victory, the size of the empire was dramatically increased. After the assassination of Caligula in AD 41, the Senate briefly considered restoring the republic, but the Praetorian Guard proclaimed Claudius emperor instead. Under Claudius, the empire invaded Britannia, its first major expansion since Augustus. After Claudius' successor, Nero, committed suicide in AD 68, the empire suffered a series of brief civil wars, as well as a concurrent major rebellion in Judea, during which four different legionary generals were proclaimed emperor. Vespasian emerged triumphant in AD 69, establishing the Flavian dynasty, before being succeeded by his son Titus, who opened the Colosseum shortly after the eruption of Mount Vesuvius. His short reign was followed by the long reign of his brother Domitian, who was eventually assassinated. The Senate then appointed the first of the Five Good Emperors. The empire reached its greatest extent under Trajan, the second in this line.

Question: Who is the mother of the person under whom the government inspiring the Carolingian empire reached its greatest extent?
Answer: Marcia