Elliptic curves and modularity
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چکیده
The discriminant Define b2 := a1 + 4a2, b4 := 2a4 + a1a3, b6 := a3 + 4a6, b8 := a1a6 + 4a2a6 − a1a3a4 + a2a3 − a4. Note that if char(K) 6= 2, then we can perform the coordinate transformation y 7→ (y− a1x− a3)/2 to arrive at an equation y = 4x + b2x +2b4x+ b6. Now for any Weierstrass equation we define its discriminant ∆ := −b2b8 − 8b4 − 27b6 + 9b2b4b6. Proposition 1. A curve C/K given by a Weierstrass equation ((1) or (2)) has ∆ 6= 0 if and only if C is smooth.
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