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§1. The Legendre Family 87

(1.3) Proposition. The orbit of λ under G acting on P 1 −{0, 1, ∞} is

1 1 λ − 1 λ
λ, 1 − λ, , , , .
λ 1 − λ λ λ − 1
If s is an element in G, then s(λ) equals one of these expressions. The curves E λ
and E λ are isomorphic, in the sense that their equations differ by a linear change of

variable which preserves the group structure, if and only if there exists s ∈ G with
s(λ) = λ .

Proof. When E λ and E λ are isomorphic in this sense, the three points of order 2 are

preserved, and, hence, the change of variable must be a composite of the two given
explicitly above. Remaining computations are left to the reader.

2
(1.4) Remark. The j-invariant j(λ) or j(E λ ) of E λ : y = x(x − 1)(x − λ) is the
value.
2
(λ − λ + 1) 3
8
j(λ) = 2 .
2
λ (λ − 1) 2
This j-invariant is a special case of the j-invariant of any cubic in normal form,
8
which was considered in detail in 3, §3, and the normalization factor 2 arises natu-
rally in the general context.

(1.5) Proposition. The j-invariant has the property that j(λ) = j(λ ) if and only if
E λ and E λ are isomorphic under changes of variable preserving the group structure.

Proof. This follows from (1.3) and the observation that j(λ) = j(λ ) if and only if
λ = s(λ) for some s ∈ G. This is a direct calculation coming from j(1 − λ) = j(λ)

and j(1/λ) = j(λ).
(1.6) Remark. The orbit of λ ∈ P 1 under G has six distinct elements, λ,1/λ,1−λ,
1/(1 − λ), λ/(λ − 1),and (λ − 1)/λ, except in these cases:
(a) j(λ) =∞ where the orbit is {0, 1, ∞}.
2
2
(b) j(λ) = 0 where the orbit is {−ρ, −ρ } for ρ + ρ + 1 = 0, i.e., ρ is a primitive
third root of unity.
3
(c) j(λ) = 12 = 1728 where the orbit is {1/2, −1, 2}.
The mapping from λ to j(λ) is shown in the following diagram with the ramifi-
cation behavior.
3
Now we point out a few things about the two exceptional values j = 12 and
j = 0.
3
1. For j = 12 take λ =−1. Then the curve is the familiar
3
2
y = x(x − 1)(x + 1) = x − x
3
2
which is one of the family y = x + ax studied in (3.2). It is closely related to
4
4
4
the Fermat curve u + v = w .