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Relative class numbers for CM fields has been computed in cases where we had both class numbers already. They are shown on the page of a CM number field (it is in the class number/class group section).
http://127.0.0.1:37777/NumberField/6.0.399424.1
http://beta.lmfdb.org/NumberField/6.0.399424.1
Here is a case where it has been computed, but the computation assumed GRH
http://127.0.0.1:37777/NumberField/42.0.2281836760183646137444154412268560109828024514076489472840222217265158917203.1
and a case where it has not been computed
http://127.0.0.1:37777/NumberField/42.0.2654972529667057457433107680535256579009544347022866599647016024950419363310098668524052547.1
If the field is not CM, we don't do anything new since it is not relevant.
It is also now a search criterion
http://127.0.0.1:37777/NumberField/
A possible future improvement is if relative_class_number is part of the search, automatically turn is_cm to true (and give a flash error if the user has set it to false).
The database nf_fields should be copied to the cloud before this goes to production.