Upvote:4
You could try submitting a Subject Access Request https://www.gov.uk/government/publications/requests-for-personal-data-uk-visas-and-immigration/request-personal-information-held-by-uk-visas-and-immigration to obtain details of the results of the document verification check https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/576713/Documentation-verification-v1.pdf that UKVI would have completed on your IELTS certificate.
Once you know more about why the document was deemed false you should be in a better position to decide how to apply again or whether to go to the expense of consulting an immigration lawyer.