  This paper classifies all modular data of integral modular fusion categories
up to rank 13. Furthermore, it also classifies all integral half-Frobenius
fusion rings up to rank 12. We find that each perfect integral modular fusion
category up to rank 13, as well as every perfect integral half-Frobenius fusion
ring up to rank 12, is trivial. We have also refined the non-pointed
odd-dimensional modular data at ranks below 25 to three items, all of rank 17,
FPdim 225, and type [[1,3],[3,8],[5,6]], filling gaps in the literature. For
rank 25, we have narrowed down the perfect case to 3 types.
  Our initial key insight is that the Egyptian fractions, which are typically
employed to list possible types, can be chosen with squared denominators. We
then develop several type criteria as initial filters. In particular, we
establish that the number of distinct basic FPdims in a non-trivial perfect
fusion ring must be at least 4. To obtain the fusion rings, we solve the
dimension and associativity equations using new features on Normaliz created
specifically for this purpose. The S-matrices (if they exist) are obtained by
self-transposing the character table, while the T-matrices are derived by
solving the Anderson-Moore-Vafa equations. Finally, we verify the extended
axioms of modular data.
  From rank 13 onward, the types were further restricted by additional
properties unique to the modular case, which involved the universal grading,
congruence representations of the modular group and Galois action, leading to
critical arithmetic constraints. In particular, we get that, up to rank 21, a
prime divisor of the global FPdim does not exceed the rank, and more strongly
up to rank 15 in the non-pointed case, does not exceed half the rank.
  Ultimately, we narrowed down the classification at rank 14 to 35 possible
types, 8 of which are non-perfect.
