Papers › Computing quadratic points on modular curves X₀(N)
Computing quadratic points on modular curves X₀(N)
Nikola Adžaga, Timo Keller, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman, Borna Vukorepa
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In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X₀(N) of genus up to 8, and genus up to 10 with N prime, for which they were previously unknown. The values of N we consider are contained in the set ℒ={58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 }. We obtain that all the non-cuspidal quadratic points on X₀(N) for N∈ℒ are CM points, except for one pair of Galois conjugate points on X₀(103) defined over ℚ(√(2885)). We also compute the j-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.
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