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For many LLL computations, Sage relies on fpLLL / fpyLLL:

• fpLLL
• fpyLLL

and on PARI/GP, see

• PARI: Vectors, matrices, linear algebra and sets: qflll
• PARI: Vectors, matrices, linear algebra and sets: qflllgram

The repositories and documentation pages linked to above might get you started on tracking what you want.