1.5.10.
The traditional narrative about Flavius Josephus and his significance as a source for Jesus’ life and for the origins of Christianity states that if Josephus had known about Jesus and the Christians,

of course he would have mentioned them. Since he did not mention them, the Christian sect was obviously so insignificant that he did not know about them or at least did not consider them worth mentioning.

The Testimonium Flavianum (Ant. 18, 3, 3) is rightly considered to be a Christian insertion. So the only historical information about Christian issues in Josephus is the mention of the stoning of the Lord’s brother James in Ant. 20, 9, 1 and the report about John the Baptist in Ant. 18, 5, 2.

1.5.11.
I advocate the following opinion on this topic: as early as under Emperor Claudius (41-54 AD) the well-known edict already existed stating that the emperor drove the Jews out of Rome because they were causing a disturbance, spurred on by a certain Chrestos (=Christ): see Suetonius, Claudius, 25, 4; compare Acts of the Apostles 18:2. We may assume that Josephus had plenty of contacts in Rome, and he must have been aware of the Roman Christian community to which Paul had already addressed an epistle.