Primitive solutions

Theorem:
Equation x^2+y^3=z^7 possesses exactly 5 non trivial (*) solutions GCD(x,y,z)=1 (we just consider x>0)

Can you find them?

Answer format: z1,z2,z3,z4,z5 sorted by ascending value
[My timing: 3,2 sec]

(*) x,y,z != 0

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Problem Maintainer :  Philippe_57721
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