Violating the Nagell-Lutz theorem


In a recent post I gave examples of elliptic curves for each of the cases that Mazur's theorem allows. One of these is particularly interesting. It's the curve

y2 + xy – 5y = x3 – 5x2

Over the rationals this has that Etors = Z2 x Z4 = <(10,20),(1,2)> = {(1,2), (10,-25), (0,5), (0,0), (-5/4,25/8), (5,0), (10,20), O}.

Note that one of these points, (-5/4,25/8), doesn't have integer coordinates. Doesn't that violate the Nagell-Lutz theorem, which tells us that torsion points need to have integer coordinates?

Not really, and here's why.

Here's one form of the Nagell-Lutz theorem:

Let y2 = x3 + ax + b be an elliptic curve over the rationals with integer coefficients and let D = 4 a3 + 27 b2. Then if P = (xP,yP) is a rational point of finite order then P has integer coordinates and either yP = 0 or yP2|D.

Note that this only applies to elliptic curves of the form E: y2 = x3 + ax + b. So because the curve in this example isn't of that form, its torsion points don't have to have integer coordainates.

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