How much does an authentic katana cost?

Could you please elaborate on the various factors that influence the cost of an authentic katana? For instance, does the craftsmanship, the materials used, the historical significance, or the artist's reputation play a significant role in determining the price? Additionally, are there any specific markers or certifications that one should look for to ensure the authenticity of a katana? Finally, could you provide an approximate range for the cost of an authentic katana, taking into account these various factors?