Legendre form
楕円曲線 “y^2 = x(x - 1)(x - λ)”

https://en.wikipedia.org/wiki/Legendre_form
Legendre form
In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because[1] the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity {\displaystyle \scriptstyle {k}}\scriptstyle {k} (the ellipse being defined parametrically by {\displaystyle \scriptstyle {x={\sqrt {1-k^{2}}}\cos(t)}}\scriptstyle{x = \sqrt{1 - k^{2}} \cos(t)}, {\displaystyle \scriptstyle {y=\sin(t)}}\scriptstyle{y = \sin(t)}).
In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric forms. A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals.
The Legendre form of an elliptic curve is given by
y^{2}=x(x-1)(x-λ)

https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf
INTER-UNIVERSAL TEICHMULLER THEORY IV: LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS
Shinichi Mochizuki April 2020
P41
Corollary 2.2. (Construction of Suitable Initial Θ-Data) Suppose that
X = P1Q is the projective line over Q, and that D ⊆ X is the divisor consisting of
the three points “0”, “1”, and “∞”. We shall regard X as the “λ-line” - i.e.,
we shall regard the standard coordinate on X = P1
Q as the “λ” in the Legendre
form “y2 = x(x-1)(x-λ)” of the Weierstrass equation defining an elliptic curve -
and hence as being equipped with a natural classifying morphism UX → (Mell)Q
[cf. the discussion preceding Proposition 1.8]. Let

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