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Thanks for computing this! I am confused though by the cubic, because that cubic is given by $$ z_{15}(z_0z_{21}-z_1z_{20})+ z_{21}(z_7z_{12}-z_0z_{15}),$$ and I think those two quadrics are part of the ideal (at least, I see them on the list). Do you have any idea what makes the cubic appear in there as well?
