Abstract
<p> In this short note, we give a method for computing a non-torsion point of smallest canonical height on a given elliptic curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E slash double-struck upper Q"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E/ \mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over all number fields of a fixed degree. We then describe data collected using this method, and investigate related conjectures of Lehmer and Lang using these data. </p>