Before Theorem 6 (in the published version) we claim that C_17 cannot appear as a torsion group over a cyclic and that this follows from M. Derickx, S. Kamienny and B. Mazur, "Rational families of 17-torsion points of elliptic curves over number fields", however this is not true. There are examples of elliptic curves over cyclic quartic fields with C_17 torsion found by Mark van Hoeij and Daeyeol Jeon inedpendently. 

This mistake has been pointed out to me by Maarten Derickx and Andreas Schweizer independently.

Here is a proof that C_17 in not a torsion group over Q(\zeta_5). 

Let K = Q(zeta_5). The prime 5 is totally ramified in K, with residue field F_5. Suppose P in Y_1(17)(K) is non-cuspidal. At the prime above 5, the corresponding elliptic curve cannot
have good reduction: otherwise its 17-torsion point would inject into an elliptic curve over F_5, but the Hasse bound gives #E(F_5) <= 10, impossible. Hence P reduces to a cusp.

Because the prime above 5 is totally ramified, all four Galois conjugates of P reduce to the same F_5-cusp c. Thus the trace divisor D = Tr_{K/Q}(P) is a rational effective divisor of degree 4 reducing to 4c. 

J_1(17)(Q) has rank 0 and no 5-torsion. Therefore reduction J_1(17)(Q) -> J_1(17)(F_5) is injective.

The eight F_5-cusps are reductions of eight rational cusps. For each of them Magma verifies

Dimension(RiemannRochSpace(4*C)) eq 1;

Hence if D reduces to 4c, then [D - 4C] reduces to 0 in the Jacobian, so injectivity gives D ~ 4C; but l(4C)=1, so D = 4C. 
That forces D to be cuspidal, contradicting that it came from a non-cuspidal P. Therefore X_1(17)(Q(zeta_5)) has no non-cuspidal points.