1728 (number)

1728 occurs in the algebraic formula for the j-invariant of an elliptic curve, as a function over a complex variable on the upper half-plane ${\displaystyle \,{\mathcal {H}}:\{\tau \in \mathbb {C} ,{\text{ }}\mathrm {Im} (\tau )>0\}}$,[19]

${\displaystyle j(\tau )=1728{\frac {g_{2}(\tau )^{3}}{\Delta (\tau )}}=1728{\frac {g_{2}(\tau )^{3}}{g_{2}(\tau )^{3}-27g_{3}(\tau )^{2}}}.}$

Inputting a value of ${\displaystyle 2i}$ for ${\displaystyle \tau }$, where ${\displaystyle i}$ is the imaginary number, yields another cubic integer:

${\displaystyle j(2i)=1728{\frac {g_{2}(2i)^{3}}{g_{2}(2i)^{3}-27g_{3}(2i)^{2}}}=66^{3}.}$

In moonshine theory, the first few terms in the Fourier q-expansion of the normalized j-invariant exapand as,[20]

${\displaystyle 1728{\text{ }}j(\tau )=1/q+744+196884q+21493760q^{2}+\cdots }$

The Griess algebra (which contains the friendly giant as its automorphism group) and all subsequent graded parts of its infinite-dimensional moonshine module hold dimensional representations whose values are the Fourier coefficients in this q-expansion.

The number of directed open knight's tours in ${\displaystyle 5\times 5}$ minichess is 1728.[21]

1728 is one less than the first taxicab or Hardy–Ramanujan number 1729, which is the smallest number that can be expressed as sums of two positive cubes in two ways.[22]