556 
[871 
871. 
A CASE OF COMPLEX MULTIPLICATION WITH IMAGINARY 
MODULUS ARISING OUT OF THE CUBIC TRANSFORMATION 
IN ELLIPTIC FUNCTIONS. 
[From the Proceedings of the London Mathematical Society, vol. xix. (1888), pp. 300, 301.] 
The case in question is referred to in my “Note on the Theory of Elliptic 
Integrals,” Math. Ann., t. xii. (1877), pp. 143—146, [657]; but I here work it out 
directly. 
y 1 + vu 2 (y + 2 u 3 ) x 2 5 
giving 
dy V v 
Vl — y 2 . 1 — v 8 y 2 V1 — X 2 . 1 
We thus have a case of complex multiplication if v 8 = u 8 , or say v = <yu, where 
7 8 = 1, or 7 denotes an eighth root of unity. Substituting in the modular equation, 
this becomes 
id (1 — y 4 ) + 2y u 2 (1 — fid) — 0, 
or, throwing out the factor u 2 and reducing, 
id — \u 2 (y 5 — 7) — y 6 = 0, 
U‘ 
7 
that is, 
| (y 4 — 1 + Vy 8 + 14y 4 + 1),