Is it cubic?

Algebra Level 4

Let f(x)f(x) be a polynomial with integer coefficients such that

f(1)=1f(2)=8f(3)=27f(5)=125f(6)=216f(7)=343. \begin{aligned} f(1) &= 1 \\ f(2) &= 8 \\ f(3) &= 27 \\ f(5) &= 125 \\ f(6) &= 216 \\ f(7) &= 343. \end{aligned}

What is the minimum possible value of f(4)?|f(4)|?


Inspiration.

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