j-invariant and the Discriminant and 12^3=1728

Welcome to Kevin John Trinder's work in progress
for the self-publishing of:

NUMBER THEORY
an unqualified demonstration June 1996

(Kevin John Trinder's reconciliation and research of,
"the theory of numbers")

ISBN 0-646-28727-3
by Author and Copyright © Kevin John Trinder

Elliptic and Modular Functions:

j-invariant and the discriminant and 1728.

and

positive integer/non-negative number 1728.

Check your outcomes with (jpeg below):

Kevin J Trinder's inverse (1/x) input - output equation

for:

non-negative numbers (x)

Notional:

harmonics

Math (mathematics) and f(x)

inverse additive, inverse groups

constant coefficient 1728,

1728

j-invariant and the discriminant

twelve cubed

12^3, 36*48

4.8 inverse .208333, 36 inverse .027777

360 inverse .002777, 1728 inverse .000578

8 inverse .125, 9 inverse .111111

72 inverse .013888, square root of three 1.73205 inverse .577350

square root of two 1.41421 inverse .707106

(square root of .5)

square root of six 2.44948 inverse .408248

inverted in position, Inverse proportion

increase proportion, decrease proportion

inverted proportion, direct opposite

inversley proportional, inverse function

.000578

mutual x times 1/x product outcome is unity.

NB 34560 (3456) is (1728 * 2 * 10) or (12^3 * 2 * 10)

NB 101.823376 is sqrt (1728 * 6) or sqrt(12^3 * 6)

Notions for me to follow up on:

trigonometric functions and their inverse, graph of the function f(x),

domain of the function f(x), composition of the function,

function f(x) and notions in common and their inverse,

the standard notion for f(x), graphs of the function,

dialating function, f(x) and zero

inverse algebraic functions, function links and the notion of inverse,

unique inverse and the notion of restricted domain,

inverse ranges and range of function, equalising the "domain" of the inverse,

inverse function with no restriction on the domain.

Elliptic modular function,

class field theory and j,

the fundamental region

algebraic definition,

the q-series and moonshine

Authors note (Kevin Trinder):

it has been said

" Two elliptic curves are isomorphic if

and

only if they have the same j-invariant.

(The reason for the constant coefficient 1728 is that j,

being dependent on the lattice periods .... and .... ,

has an explicit formula

in terms of them out of which

1728 falls out naturally.) "

Now see my jpeg notation below

for further explanation of

positive integer 1728.

Transforming a non-negative number

(positive number x)

into a positive inverse value 1/x: