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.


-------------------------------------------------------------------------------
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
-------------------------------------------------------------------------------


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
n1n2n3UABCD
-2 -2 -1 11225395/111212249 47886740272114976 8813425670440240 56827813308111785 62940516903410601
-1 -2 -1 30671/229738 2448718655 664793200 3134081336 3393603777
-1 -2 0 129559319/270196535 3579087147375440 14890026433468471 18565945114216720 20249506709579721
-1 -1 -1 18247/19530 260052385 582665296 186668000 589845921
-1 -1 0 14392647/14506151 2480452675600 4987588419655 502038853976 5062297699257
0 -1 -1 -31/467 18796760 2682440 15365639 20615673
0 -1 0 -3015/9707 630662624 275156240 219076465 638523249
0 0 -1 -3015/9707 630662624 275156240 219076465 638523249
1 0 -1 14392647/14506151 2480452675600 4987588419655 502038853976 5062297699257
1 0 0 18247/19530 260052385 582665296 186668000 589845921
1 1 -1 129559319/270196535 3579087147375440 14890026433468471 18565945114216720 20249506709579721
1 1 0 30671/229738 2448718655 664793200 3134081336 3393603777
2 1 0 11225395/111212249 47886740272114976 8813425670440240 56827813308111785 62940516903410601



[1] Noam D Elkies: On A^4+B^4+C^4=D^4,Mathematics of Computation,Oct.1988