A table of elliptic curves E/Q with III(E/Q) containing a copy of (Z/3Z)^4

51870 br 5 [1,0,1,-2550136823,-49567293320614] 0 2 81
51870 br 6 [1,0,1,-40802189303,-3172296548756902] 0 2 81
53466 t 3 [1,0,0,-132358428,-586116148734] 0 1 81
106210 n 3 [1,0,0,-192492345,-1027957325155] 0 1 81
123690 bp 3 [1,0,0,-12395676,-39524634594]      0       1       81
128898 cf 3 [1,-1,1,-306044246,-2060669372057]  0       1       81
166530 ba 3 [1,0,1,-55220133,-157945382384]     0       1       81
186186 i 3 [1,0,0,-16620758,-26082391944]	0	1	81
190190 u 3 [1,0,0,-260158021,-1615138515149]	0	1	81
211302 b 3 [1,-1,1,-591553289,-5537673231713]	0	1	81

These elliptic curves may be written as
      C_la  :  y^2 + xy + la y = x^3 - 5 la x - la (7 la + 1)
where
      la = t (t^2 + t + 1) / (3 (2t + 1)^3) 
and 
      t = 135/71, 225/233, 54/55, 60/59, -54/71, 22/23, 45/41, 36/25, 66/61, 26/27.

Further examples of this form may easily be constructed using Theorem 4 in

T. A. Fisher, The Cassels-Tate pairing and the Platonic solids, 
J. Number Theory 98 (2003) 105-155.