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- W165382684 abstract "The purpose of this article is to describe a relationship between several cubic polynomials and elliptic curves, and show a clearer view on it than that in the former half of our previous work [Mi-2003]. For a monic irreducible cubic polynomial P(u) in u over ℚ, the curve E = E(P(u)) defined by the equation w3 = P(u) is an elliptic curve whose j-invariant is equal to 0. We describe the set E[ℚ] of all rational points of E over ℚ by use of a root ξ of P(u) as $$ mathcal{W}left( xi right) = left{ {alpha = qxi + rleft| {N_{K/mathbb{Q}} } right.left( alpha right) = 1,q,r in mathbb{Q}} right}. $$ Then we show that the short form of E is a Mordell curve, y2 = x3 + k, with a certain rational number k determined by the coefficients of P(u). It is also pointed out that E(P(u)) is essentially dependent on the polynomial P(u) rather than the cubic field ℚ(ξ) even though E[ℚ] is completely described by the subset W(ξ) of the cubic field." @default.
- W165382684 created "2016-06-24" @default.
- W165382684 creator A5070628085 @default.
- W165382684 date "2006-06-02" @default.
- W165382684 modified "2023-09-27" @default.
- W165382684 title "Cubic Fields and Mordell Curves" @default.
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- W165382684 doi "https://doi.org/10.1007/0-387-30829-6_12" @default.
- W165382684 hasPublicationYear "2006" @default.
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