  This paper investigates a specific smooth quintic surface suggested by Barth
for it contains the current record of 75 lines over the complex numbers. Our
main incentive is to prove that the complex quintic has Picard number 41, and
to compute the Neron-Severi group up to a 2-power index. We also compute Picard
numbers for reductions to positive characteristic and verify the Tate
conjecture.
