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<p> <span>We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quadratic fields </span><span>Q(m)</span><span> for </span><span>m</span><span style="font-family:宋体;">≤</span><span>200</span><span>. These results of computations give best-possible data including structures of Mordell-Weil groups over some real quadratic fields via two-descent.</span><span> </span><span>We also prove similar results for the case of certain cubic fields. Especially, we give the first example of elliptic curve having everywhere good reduction over a pure cubic field using our method.</span><span></span> </p>

Estimating the number of pairing-friendly elliptic curves is important for obtaining such a curve with a suitable security level and high efficiency. For 128-bit security level, M. Naehrig and J. Boxall estimated the number of Barreto-Naehrig (BN) curves. For future use, we extend their results to higher security levels, that is, to count Kachisa-Schaefer-Scott (KSS) curves with 192-and 224-bit security levels. Our efficient counting is based on a number-theoretic conjecture, called the Bateman-Horn conjecture. We verify the validity of using the conjecture and confirm that an enough amount of KSS curves can be obtained for practical use.