Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with b=-1 and c=-a-d. Parameters (a,d)=(-4/5,15/4) and t=d=15/4 give w=(d+1)(a^2+ad+d^2)(d^2-d+1)=16097959/25600 on w^2=D_{a,d}(t), where D_{a,d}(t)=L_{a,d}(t)^2-4(a^3+1)d^3(t-a)(t-d)^3, L_{a,d}(t)=(d^3+1)t^2+(3a^2d^2+ad^3-2a-3d)t-2a^2d^3+a^2+3ad+3d^2. Cubic normalization of k=(a^3+1)(t-a)w/8 gives u^3+v^3=89359770409. PARI/GP 2.17.4 ellrank bounds [7,7].
jesper-petersen ·