Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-5,-1,1,4) and specialization t=-5/4. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=15506820. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].
jesper-petersen ·