I made the table of the known solutions for A4+B4+C4=D4.
First,We transform the equation (1) to the elliptic curve (4).
Next,We look for the generators of (4) by using Cremona's mwrank.
And,We can get the several small solutions by generators.
A4+B4+C4=D4...................................(1)
Set r=A/D,s=B/D,t=C/D,then
r4+s4+t4=1....................................(2)
Equation (1) is finally transformed to the next equation (3).
By Elkies's paper([1].Elkies),Set (m,n)=(20,-9) then we obtain
V2=-19435071440U4+5351620404U3-130338882000U2+194951575764U+357457601448.......(3)
If we obtain a rational solution of equation (3),we can get a solution of (1)
with the next process.
k=U
x=1/318(43169k2-657351-121578k)/(881k2+4083)
y=-1/106(20263k2-93909+285806k)/(881k2+4083)
r=x+y=A/D,s=x-y=B/D
Next, we must get the rational solutions U of (3).
First,we transform (3) to Minimal Weierstrass form (4),and get the rational solutions of (4).
After that,by using group law of generators, we can get infinitely many rational solutions of (4).
Minimal Weierstrass form: y2 = x3+2265722465761x-3154189403034549278.............(4)
We could not get the generators of (4) by Cremona's mwrank.
But, We got three rational points p1,p2,p3 on the curve (4).
I found several solutions under the condition of D<10^20.
(n1,n2,n3)=(0,1,0):MacLeod's case
(n1,n2,n3)=(1,0,0):Fly's case
p1=[978559, 0]
p2=[47971729/49, 16603172706/343]
p3=[1237921, 1244044242]
p=n1*p1+n2*p2+n3*p3
A4+B4+C4=D4