For every non-CM elliptic curve over Q, the index of the adelic Galois image is bounded by 10^21(h_F(E)+40)^4.42, and by h_F(E)^{3+o(1)} as the height grows.
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Effective bounds for adelic Galois representations attached to elliptic curves over the rationals
For every non-CM elliptic curve over Q, the index of the adelic Galois image is bounded by 10^21(h_F(E)+40)^4.42, and by h_F(E)^{3+o(1)} as the height grows.