JohnCremona
commented
5 years ago Bianchi newforms with no elliptic curve (or "false elliptic curves", genuine or not)
www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/34225.7/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/34225.7/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/34225.3/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/34225.3/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/5625.3/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/5625.1/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/6561.5/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/6561.5/d www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/21609.3/c www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/21609.3/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/21609.1/c www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/21609.1/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/67081.3/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/67081.3/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/67081.7/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/67081.7/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/61009.1/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/61009.1/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/61009.9/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/61009.9/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/123201.1/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/123201.1/c www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/123201.3/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/123201.3/c www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/5625.1/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/6561.1/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/30625.3/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/30625.1/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/50625.1/d www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/50625.1/c www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/65536.1/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/65536.1/e www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/104976.1/a www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/10000.1/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/10000.5/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/30625.1/e www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/30625.1/c www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/30625.1/d www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/40000.7/b www.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/40000.1/b
Of these, 24 (which are base-change up to twist) correctly say "no elliptic curve", while 16 (conductor norm 34225 over 2.0.4.1, 67081 and 61009 and 123201 over 2.0.3.1) misleadingly say "isogeny class
AndrewVSutherland
The table bmf_forms contains 185581 newforms of dimension 1 over the 5 fields 2.0.d.1 for d in {1,8,3,7,11}. This includes 185541 with associated elliptic curves and 40 (listed below) where is no associated curve. these are not just missing, they do not exist (sometimes these are known as "fake elliptic curves"); instead there is an abelian surface whose L-function which matches that of the elliptic curve.
Over these fields the table ec_nfcurves contains 186507 isogeny classes of elliptic curves, including 966 classes whose label includes "CM", which have CM by an order in the base field, which are not associated to cuspidal BMFs. That leaves 185541 classes which match bijectively to the BMFs of the preceding paragraph.
The reason for this issue is (1) to record these numbers (as of 2019-05-08), (2) to list the exceptional BMFs, and (3) to put into the agenda the need to indicate on the home pages of the 40 exceptional BMFs that they are exceptional. Currently their home pages just say, in the Friends box, "No elliptic curve" or (e.g.) "Isogeny class 2.0.3.1-67081.3-a missing" depending on the type of exception, not all of which are currently detected automatically.