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Exponential for formal group of elliptic curve

asked 2020-03-17 12:58:09 -0500


This is a short and potentially quite simple question - the formal group object for an elliptic curve allows you to compute its logarithm, i.e. the isomorphism from the formal group to the additive formal group. Is it possible to find the exponential as well? If this is not built in, this would just reduce to finding a power series $g$ such that $f(g(T))=T$ for a given $f$. Is this possible with the power series tools?


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answered 2020-03-18 07:09:37 -0500

FrédéricC gravatar image

You can use .reverse:

sage: x = PowerSeriesRing(QQ, 'x').gen()
sage: y = exp(x) - 1
sage: y.reverse()
x - 1/2*x^2 + 1/3*x^3 - 1/4*x^4 + 1/5*x^5 - 1/6*x^6 + 1/7*x^7 - 1/8*x^8 + 1/9*x^9 - 1/10*x^10 + 1/11*x^11 - 1/12*x^12 + 1/13*x^13 - 1/14*x^14 + 1/15*x^15 - 1/16*x^16 + 1/17*x^17 - 1/18*x^18 + 1/19*x^19 + O(x^20)
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Asked: 2020-03-17 12:58:09 -0500

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Last updated: Mar 18