Elliptic curve definition

Number Theory Level 3

Consider the curves given by the following three equations.

(1) y 2 = − 1 16 x 2 + 1 y^2 = -\dfrac 1{16}x^2 + 1

(2) y 2 = x 3 − 3 x + 2 y^2 = x^3-3x+2

(3) y 2 = x 3 − 3 x y^2 = x^3-3x

Which of these curves are elliptic curves ?

(2) and (3) (1), (2), and (3) (3) only (1) and (2)

Patrick Corn by Patrick Corn , 44, USA

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1 solution

Eli Ross Staff
May 2, 2016

Relevant wiki: Elliptic Curves

An elliptic curve is of the form y 2 = f ( x ) y^2 = f(x) where f ( x ) f(x) is a cubic polynomial with no repeated roots . So, (1) is out, since f ( x ) f(x) is quadratic.

(II) and (III) look more promising. However, x 3 − 3 x + 2 = ( x − 1 ) ( x 2 + x − 2 ) = ( x − 1 ) 2 ( x + 2 ) x^3 - 3x + 2 = (x-1)(x^2 +x- 2) = (x-1)^2 (x+2) has a double root at x = 1 , x=1, so (II) is not an elliptic curve.

(III) checks out, though, since x 3 − 3 x = x ( x 2 − 3 ) , x^3 -3x = x(x^2-3), which has distinct roots of 0 , 3 , − 3 . 0,\sqrt{3},-\sqrt{3}.

16 Helpful 2 Interesting 3 Brilliant 1 Confused

Does "no repeated roots" means "no multiplicity roots"?

Leonardo Vannini - 5 years ago

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Yes - no roots with multiplicity greater than 1.

Eli Ross Staff - 5 years ago

(ii) can also be given as x − 1 ) ( x + 1 ) ( x 2 − 1 ) ( x 2 + 1 ) . . . x-1)(x+1)(x^2-1)(x^2+1)... in an infinite sequence.

Ariijit Dey - 3 years ago

You may also check the derivative to detect multiple roots. For (iii): f ( 1 ) = 0 , f ( 1 ) ( 1 ) = 0 , f ( 2 ) ( 1 ) = 6 f(1)=0,\:\: f^{(1)}(1)=0,\:\: f^{(2)}(1)=6 , then 1 is a zero of multiplicity 2!

Carsten Meyer - 1 year, 1 month ago

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