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72 3. Elliptic Curves and Their Isomorphisms

3
(3.5) Remarks. The polynomial f (x) = x + px + q ∈ k[x] has a repeated root
in some extension field of k if and only if D( f ) = 0. For a field k of characteristic
2
unequal to 2 the cubic curve given by the equation y = f (x) is nonsingular if
and only if D( f ) = 0. This was already remarked in 1(2.1), and it is related to the

calculation 2yy = f (x) showing that there is a well-defined tangent line if and only
if f (x) and f (x) do not have a common solution See also (8.2).

Return now to
3
2
(N 2 ) y = x − c 4 x − c 6 = f (x),
48 864
4
3
5
so that p =−c 4 /48 and q =−c 6 /864. Using 864 = 2 · 3 and 48 = 2 · 3, we see
that
3
c − c 2
4 4 6
−2 D( f ) = 3 = .
12
In conclusion we have the following result in characteristic different from 2 and 3.
(3.6) Proposition. Over a field k of characteristic different from 2 or 3, the cubic
equation
c 4 c 6
2 3
y = x − x −
48 864
represents an elliptic curve if and only if = 0. Also ω − dx/2y.

3
(3.7) Remark. For j = 0or 12 the following cubic
36 1
2 3
y + xy = x − x −
j − 1728 j − 1728
defines an elliptic curve with j-invariant equal to j over any field k. This is a straight-
forward calculation which is left to the reader to verify. The elliptic curve with equa-
2
2
3
3
tion y = x + a has j = 0, and the elliptic curve with equation y = x + ax has
3
j = 12 = 1728.
This topic is also taken up in §8.
Exercises

2
3
1. Derive the formula for the discriminant of the cubic polynomial f (x) = x +ax +bx+c.
2. Derive the formula for the discriminant of the cubic polynomial
f (x) = (x − α 1 )(x − α 2 )(x − α 3 ).
3. Derive the formula for the discriminant of the quartic polynomial
4 2
f (x) = x + ax + bx + c.