Editing Weierstrass elliptic function (section)

===Modular discriminant===
[[Image:Discriminant real part.jpeg|thumb|The real part of the discriminant as a function of the square of the nome ''q'' on the unit disk.]]

The ''modular discriminant'' <math>\Delta</math> is defined as the [[discriminant]] of the characteristic polynomial of the differential equation <math display="block"> \wp'^2(z) = 4\wp ^3(z)-g_2\wp(z)-g_3</math> as follows:
<math display="block"> \Delta=g_2^3-27g_3^2. </math>
The discriminant is a modular form of weight <math>12</math>. That is, under the action of the [[modular group]], it transforms as
<math display="block">\Delta \left( \frac {a\tau+b} {c\tau+d}\right) = \left(c\tau+d\right)^{12} \Delta(\tau) </math>
where <math>a,b,d,c\in\mathbb{Z}</math> with <math>ad-bc = 1</math>.<ref>{{Cite book|last=Apostol | first = Tom M.| url=https://www.worldcat.org/oclc/2121639|title=Modular functions and Dirichlet series in number theory| date=1976| publisher=Springer-Verlag|isbn=0-387-90185-X|location=New York|pages=50|oclc=2121639}}</ref>

Note that <math>\Delta=(2\pi)^{12}\eta^{24}</math> where <math>\eta</math> is the [[Dedekind eta function]].<ref>{{Cite book| last=Chandrasekharan, K. (Komaravolu), 1920-|url=https://www.worldcat.org/oclc/12053023|title=Elliptic functions| date=1985| publisher=Springer-Verlag|isbn=0-387-15295-4|location=Berlin|pages=122|oclc=12053023}}</ref>

For the Fourier coefficients of <math>\Delta</math>, see [[Ramanujan tau function]].