For rational CM elliptic curves, the torsion growth over quadratic fields is fully described by the CM invariants (cm, k): the possible growth groups are C3, C4, C6, C2 x C2, C2 x C4, C2 x C6 and C3 x C3, and the fields realizing each growth are listed explicitly.
Chou, Torsion of rational elliptic curves over quartic Galois num ber fields
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Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields
For rational CM elliptic curves, the torsion growth over quadratic fields is fully described by the CM invariants (cm, k): the possible growth groups are C3, C4, C6, C2 x C2, C2 x C4, C2 x C6 and C3 x C3, and the fields realizing each growth are listed explicitly.