Question: Which Roman emperor was personally known by or written about by both Thrasyllus of Mendes and Tacitus?

full_context: Document 1: Titus Prifernius Geminus: Titus Prifernius Geminus (full name Titus Prifernius Paetus Rosianus Geminus) was a Roman senator who lived in the second century. He is best known as a friend and correspondent of Pliny the Younger, who addresses him as "Geminus"; he served as quaestor to Pliny for the latter's consulship in AD 100, and five letters Pliny wrote to Geminus have survived. Although the letters convey a genuine friendship between the two (VII.1 mentions Geminus' illness), the first one appears only in the latter books of Pliny's collection; Ronald Syme explains this may be due to the fact that he, like Quintus Corellius Rufus and Calestrius Tiro were living in Rome at the same time as him.
Document 2: Yahya Ibn al-Batriq: Abu Yahya Ibn al-Batriq (working 796 - 806) was a Syrian scholar who pioneered the translation of ancient Greek texts into Arabic, a major early figure in the transmission of the Classics at the close of Late Antiquity. He translated for Al-Mansur the major medical works of Galen and Hippocrates, and also translated Ptolemy's "Tetrabiblos".
Document 3: Claudia Capitolina: Claudia Capitolina (Greek: η Κλαuδία Καπιτωλίνα ) was an Egyptian Greek woman who lived in the Roman Empire, in the 1st century and possibly in the 2nd century. Capitolina came from a distinguished family of Equestrian rank. She was born and raised in Alexandria, Egypt. Capitolina was the daughter and only child of Tiberius Claudius Balbilus by his unnamed wife. The cognomen "Capitolina", is probably from her maternal side. Her father was one of the highest magistrates of Equestrian rank that served in Rome. Balbilus was an astrologer and a learned scholar, who was later Prefect of Egypt. Capitolina’s paternal grandfather, was an Egyptian Greek Grammarian and Astrologer called Thrasyllus of Mendes or Tiberius Claudius Thrasyllus, who was a friend of the Roman Emperor Tiberius, while her paternal grandmother was Greek Princess Aka II of Commagene, who was a great, granddaughter of King Antiochus I Theos of Commagene. Her paternal cousin was Ennia Thrasylla who married the Praetorian prefect of the Praetorian Guard, Naevius Sutorius Macro.
Document 4: Claudius: Claudius ( ; Latin: "Tiberius Claudius Caesar Augustus Germanicus" ; 1 August 10 BC – 13 October 54 AD) was Roman emperor from 41 to 54. A member of the Julio-Claudian dynasty, he was the son of Drusus and Antonia Minor. He was born at Lugdunum in Gaul, the first (and until Trajan, the only) Roman Emperor to be born outside Italy. Because he was afflicted with a limp and slight deafness due to sickness at a young age, his family ostracized him and excluded him from public office until his consulship, shared with his nephew Caligula in 37.
Document 5: Domitian: Domitian ( ; Latin: "Titus Flavius Caesar Domitianus Augustus" ; 24 October 51 – 18 September 96) was Roman emperor from 81 to 96. He was the younger brother of Titus and son of Vespasian, his two predecessors on the throne, and the last member of the Flavian dynasty. During his reign, his authoritarian rule put him at sharp odds with the senate, whose powers he drastically curtailed.
Document 6: History of the Roman Empire: The history of the Roman Empire covers the history of Ancient Rome from the fall of the Roman Republic in 27 BC until the abdication of the last Western emperor in 476 AD. Rome had begun expanding shortly after the founding of the Republic in the 6th century BC, though it did not expand outside of the Italian Peninsula until the 3rd century BC. Civil war engulfed the Roman state in the mid 1st century BC, first between Julius Caesar and Pompey, and finally between Octavian and Mark Antony. Antony was defeated at the Battle of Actium in 31 BC. In 27 BC the Senate and People of Rome made Octavian "imperator" ("commander") thus beginning the Principate, the first epoch of Roman imperial history usually dated from 27 BC to 284 AD; they later awarded him the name Augustus, "the venerated". The success of Augustus in establishing principles of dynastic succession was limited by his outliving a number of talented potential heirs: the Julio-Claudian dynasty lasted for four more emperors—Tiberius, Caligula, Claudius, and Nero—before it yielded in 69 AD to the strife-torn Year of Four Emperors, from which Vespasian emerged as victor. Vespasian became the founder of the brief Flavian dynasty, to be followed by the Nerva–Antonine dynasty which produced the "Five Good Emperors": Nerva, Trajan, Hadrian, Antoninus Pius and the philosophically inclined Marcus Aurelius. In the view of the Greek historian Dio Cassius, a contemporary observer, the accession of the emperor Commodus in 180 AD marked the descent "from a kingdom of gold to one of rust and iron"—a famous comment which has led some historians, notably Edward Gibbon, to take Commodus' reign as the beginning of the decline of the Roman Empire.
Document 7: Tacitus (emperor): Tacitus ( ; Latin: "Marcus Claudius Tacitus Augustus" ; c. 200 – June 276), was Roman Emperor from 275 to 276. During his short reign he campaigned against the Goths and the Heruli, for which he received the title "Gothicus Maximus".
Document 8: Aulus Cremutius Cordus: Aulus Cremutius Cordus (died 25 AD) was a Roman historian. There are very few remaining fragments of his work, principally covering the civil war and the reign of Augustus Caesar. In 25 AD he was forced by Sejanus, who was praetorian prefect under Tiberius, to take his life after being accused of "maiestas (treason)". He had been accused by Satrius Secundus of having eulogized Brutus and spoken of Cassius as the last of the Romans, which was considered an offence under the "lex majestatis," and the senate ordered the burning of his writings. Seneca the Younger, however, tells us that he most likely incurred Sejanus' displeasure for criticising him, because Sejanus had commissioned a statue of himself. We also know from this source—a letter to Cordus' daughter Marcia—that he starved himself to death. She was also instrumental in saving his work, so that it could be published again under Caligula. Apart from Seneca, he is mentioned by Tacitus, Quintilian, Suetonius and Dio Cassius. Even though Cordus committed suicide, his work survived, prompting Tacitus to deride "the stupidity of people who believe that today's authority can destroy tomorrow's memories."
Document 9: Tetrabiblos: Tetrabiblos (Τετράβιβλος ) 'four books', also known in Greek as Apotelesmatiká (Ἀποτελεσματικά ) "Effects", and in Latin as Quadripartitum "Four Parts", is a text on the philosophy and practice of astrology, written in the 2nd century AD by the Alexandrian scholar Claudius Ptolemy ( AD 90– AD 168).
Document 10: Thrasyllus of Mendes: Thrasyllus of Mendes ( ; Greek: Θράσυλλος Μενδήσιος ), also known as Thrasyllus of Alexandria and by his Roman citizenship name Tiberius Claudius Thrasyllus ( Τιβέριος Κλαύδιος Θράσυλλος ; fl. second half of the 1st century BC and first half of the 1st century – died 36,), was an Egyptian Greek grammarian and literary commentator. Thrasyllus was an astrologer and a personal friend of the Roman emperor Tiberius, as mentioned in the "Annals" by Tacitus and "The Twelve Caesars" by Suetonius.
Document 11: Trajan: Trajan ( ; Latin: "Imperator Caesar Nerva Traianus Divi Nervae filius Augustus" ; 18 September 538August 117 AD) was Roman emperor from 98 to 117AD. Officially declared by the Senate "optimus princeps" ("the best ruler"), Trajan is remembered as a successful soldier-emperor who presided over the greatest military expansion in Roman history, leading the empire to attain its maximum territorial extent by the time of his death. He is also known for his philanthropic rule, overseeing extensive public building programs and implementing social welfare policies, which earned him his enduring reputation as the second of the Five Good Emperors who presided over an era of peace and prosperity in the Mediterranean world.
Document 12: Agricola (book): The Agricola (Latin: "De vita et moribus Iulii Agricolae" , lit. On the life and character of Julius Agricola) is a book by the Roman historian Tacitus, written  AD 98 , which recounts the life of his father-in-law Gnaeus Julius Agricola, an eminent Roman general and Governor of Britain from AD 77/78 - 83/84. It also covers, briefly, the geography and ethnography of ancient Britain. As in the "Germania", Tacitus favorably contrasts the liberty of the native Britons to the corruption and tyranny of the Empire; the book also contains eloquent and forceful polemics against the rapacity and greed of Rome.
Document 13: Tacitus: Publius (or Gaius) Cornelius Tacitus ( ; ] ;  56 AD –  120 AD ) was a senator and a historian of the Roman Empire. The surviving portions of his two major works—the "Annals" and the "Histories"—examine the reigns of the Roman emperors Tiberius, Claudius, Nero, and those who reigned in the Year of the Four Emperors (69 AD). These two works span the history of the Roman Empire from the death of Augustus, in 14 AD, to the years of the First Jewish–Roman War, in 70 AD. There are substantial lacunae in the surviving texts, including a gap in the "Annals" that is four books long.
Document 14: Heliodorus (philosopher): Heliodorus (Greek: Ἡλιόδωρος ) is cited as the author of a work titled "Commentary" (dated 564 AD), which has been preserved, on the "Introduction" or "Rudiments" of Paulus Alexandrinus, the 4th century Alexandrian astrologer. The name "Heliodorus" appears only on the later of two groups of manuscripts, and so is somewhat doubtful. Leendert Westerink has argued that the commentary consists of notes of lectures, most likely given by the 6th-century philosopher and astrologer, Olympiodorus, in 564 AD. The Greek text of his commentary on Aristotle's "Nicomachean Ethics" has been published in vol. 19.2 of "Commentaria in Aristotelem Graeca" (CAG).
Document 15: Maximus of Tyre: Maximus of Tyre (Greek: Μάξιμος Τύριος ; fl. late 2nd century AD), also known as Cassius Maximus Tyrius, was a Greek rhetorician and philosopher who lived in the time of the Antonines and Commodus, and who belongs to the trend of the Second Sophistic. His writings contain many allusions to the history of Greece, while there is little reference to Rome; hence it is inferred that he lived longer in Greece, perhaps as a professor at Athens. Although nominally a Platonist, he is really an Eclectic and one of the precursors of Neoplatonism.
ground_truth: Tiberius