Solving the nome 
for the parameter 
gives
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is a Jacobi theta function, 
is the Dedekind
eta function, and 
is the nome.
The inverse nome function is essentially the same as the elliptic lambda function, the difference being that elliptic lambda function is a function
of the half-period ratio 
, while the inverse nome is a function of the nome 
, where 
is itself a function of 
.
The inverse nome is implemented as InverseEllipticNomeQ[q] in the Wolfram Language.
As a rule, inverse and direct functions satisfy the relation 
-for example, 
. The inverse nome is an exception to this
rule due to a historical mistake made more a century ago. In particular, the inverse
nome and nome itself are connected by the opposite relation
.
Special values include
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
although strictly speaking, 
is not defined at 1 because 
is a modular function, therefore has a dense set of
singularities on the unit circle, and is therefore only defined strictly inside the
unit circle.
It has series
 |
(6)
|
(OEIS A115977).
It satisfies
 |
(7)
|
