In most of the current schemes, it is very unlikely that elliptic curves are defined over F_1.
They are certainly not in Deitmar's or Toen-Vaquie since they restric to toric varieties. For Soule/Connes-Consani old notions, all the examples found so far seem to come from torified varieties (as defined by Oliver and myself in this paper. Also in CC new notion, up to the torsion part of the monoidal scheme their schemes appear to be generalized torified (see Theorem 2.2 in the reference you mention). But all torified schemes are rational, so elliptic curves are not in there.
On the other hand, Manin was very confident that the set of torsion points of an elliptic curve would define a convenient model for it. But if that was the case I don't think it would fit CC models, but rather Manin/Marcolli analytic approach, conjecturally related to Borger's, but no there is clear explanation on that relation just yet.
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