36 2. Elliptic curves Example 2.6.2. Sometimes the reduction of a model for an elliptic curve E modulo a prime p is not smooth, but it is smooth for some other models of E. For instance, consider the curve E : y2 = x3 + 15625. Then E E mod 5 is not smooth over F5 because the point (0,0) mod 5 is a singular point. However, using the invertible change of variables (x, y) (52X,53Y ), we obtain a new model over Q for E given by E : Y 2 = X3 + 1, which is smooth when we reduce it modulo 5. The problem here is that the model we chose for E is not minimal. We describe what we mean by minimal next. Definition 2.6.3. Let E be an elliptic curve given by y2 = x3 +Ax+ B, with A, B Q. (1) We define ΔE, the discriminant of E, by ΔE = −16(4A3 + 27B2). For a definition of the discriminant for more general Weier- strass equations, see for example [Sil86], p. 46. (2) Let S be the set of all elliptic curves E that are isomor- phic to E over Q (see Definition 2.2.4) and such that the discriminant of E is an integer. The minimal discriminant of E is the integer ΔE that attains the minimum of the set {|ΔE | : E S}. In other words, the minimal discriminant is the smallest integral discriminant (in absolute value) of an elliptic curve that is isomorphic to E over Q. If E is the model for E with minimal discriminant, we say that E is a minimal model for E. Example 2.6.4. The curve E : y2 = x3 + 56 has discriminant ΔE = −2433512, and the curve E : y2 = x3 + 1 has discriminant ΔE = −2433. Since E and E are isomorphic (see Definition 2.2.4 and Example 2.6.2), then ΔE cannot be the minimal discriminant for E and y2 = x3 + 56 is not a minimal model. In fact, the minimal discriminant is ΔE = −432 and E is a minimal model. Before we go on to describe the types of reduction modulo p that one can encounter, we need a little bit of background on types of singularities. Let E be a cubic curve over a field K with Weierstrass
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