What does the amount on line 11 of 1040 come from? It appears to come from a computation of some sort of minimum tax.

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It's complicated. See the instructions for line 11 in the IRS instructions for Form 1040, beginning with "Line 11a" on page 38. Schedule 2 might also be involved. The instructions for Schedule 2 are on page 98. You can download the Form 1040 instructions from the following link.

https://www.irs.gov/pub/irs-pdf/i1040gi.pdf

One of the reasons you use TurboTax is so that you don't have to wade through pages of complicated instructions like this.