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)=(8,-5) then we obtain
V2=-31790U4+36941U3-56158U2+28849U+22030.......(3)
If we obtain a rational solution of equation (3),we can get a solution of (1)
with the next process.
k=U*7-2
x=(51k2-34k-5221)/(14(17k2+779))
y=(17k2+7558k-779)/(42(17k2+779))
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: y^2 = x^3-x^2+2815805388x-94443526967868.............(4)
We can get the generators of (4) by Cremona's mwrank.
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Rank = 3
After descent, rank of points found is 3
Generator 1 is [40984562244 : 7472357686746 : 1225043]; height 13.0483941769441
Generator 2 is [7682852 : 2843134723 : 64]; height 11.1574810989791
Generator 3 is [287246048321418303786 : 267958777554236828867184 : 3312733367320
57]; height 28.1492226634143
The rank has been determined unconditionally.
The basis given is for a subgroup of full rank of the Mordell-Weil group
(modulo torsion), possibly of index greater than 1.
Regulator (of this subgroup) = 1267.61149551003
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We got three generators p1,p2,p3 of (4) by Cremona's mwrank.
I found several solutions under the condition of D<10^20.
p1=[[383033292/11449,7472357686746/1225043]
p2=[1920713/16,2843134723/64]
p3=[4151374392227802/4787671249,267958777554236828867184/331273336732057]
p=n1*p1+n2*p2+n3*p3
A4+B4+C4=D4