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The authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions W_Q on X₀(N) for N ≤ 50 are Weierstrass points or not.

The design is by using Lawittes's and Schoeneberg's theorems.

Finding all Weierstrass points on X₀(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.

The Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X₀(N).

Let E be an elliptic curve with Weierstrass form y2=x3−px, where p is a prime number and let E[m] be its m-torsion subgroup. Let p1=(x1,y1) and p2=(x2,y2) be a basis for E[m], then we prove that ℚ(E[m])=ℚ(x1,x2,ξm,y1) in general. We also find all the generators and degrees of the extensions ℚ(E[m])/ℚ for m=3 and m=4.