Calculating $[7](t)$ polynomial of elliptic curve with computer

algebraic-number-theorycomputational mathematicselliptic-curvessagemath

Let $E:y^2=x^3+x/\Bbb{Q}_7(\sqrt{-1})$ be an elliptic curve. Let $ \hat E$ be a formal group of $E$.
I want to prove $[7](t)≡t^{49}mod7\Bbb{Z}_7$ does not hold, so indeed I want to know the coeffients of $t^{49},t^{50},t^{51},・・,$ but this is impossible with hand calculation.
But I'm not familiar with computer calculating. I want to be able to use computer, but result only is ok here, thank you for your help.

Best Answer

Working over $\Bbb F_7$, the field with seven elements, sage gives:

sage: EllipticCurve(GF(7), [1, 0]).formal_group().mult_by_n(7, 344)
6*t^49 + O(t^344)

Over $\Bbb Q$ the result is not so simply displayed, i will adjust manually:

sage: EllipticCurve(QQ, [1, 0]).formal_group().mult_by_n(7, 63)
7*t - 6720*t^5 + 5367040*t^9 - 4353090560*t^13 + 3530994503680*t^17 
    - 2864078396211200*t^21
    + 2323127369478766592*t^25
    - 1884348184548019011584*t^29
    + 1528443132789675420286976*t^33 
    - 1239759418841491638287597568*t^37
    + 1005600655750474908385232814080*t^41
    - 815668478478403789879107278340096*t^45
    + 661609619065682693232195492602970112*t^49
    - 536648527667515381010030082244189618176*t^53 
    + 435289382059484601080905836042899144310784*t^57
    - 353074379906097040686068218826179876382310400*t^61 + O(t^63)

And indeed, the coefficient in $t^{49}$ is $-1$ modulo seven:

sage: 661609619065682693232195492602970112 % 7
6

Later comment: One can reproduce the same also in pari/gp. See also:

MO related question

Code:

{ E = ellinit([0, 0, 0, 1, 0]); 
  prec = 70; 
  f = subst(ellformalexp(E, prec), x, 7*ellformallog(E, prec));
  }

And after copy+paste into the pari interpreter:

? f
%13 = 7*x - 6720*x^5 + 5367040*x^9 - 4353090560*x^13 + 3530994503680*x^17
          - 2864078396211200*x^21
          + 2323127369478766592*x^25
          - 1884348184548019011584*x^29
          + 1528443132789675420286976*x^33
          - 1239759418841491638287597568*x^37
          + 1005600655750474908385232814080*x^41
          - 815668478478403789879107278340096*x^45
          + 661609619065682693232195492602970112*x^49
          - 536648527667515381010030082244189618176*x^53
          + 435289382059484601080905836042899144310784*x^57
          - 353074379906097040686068218826179876382310400*x^61
          + 286387683421690457469675756519873962955287560192*x^65
          - 232296393857452115655956549775515957596755312246784*x^69 + O(x^71)

And pari gives the result almost instantly. (In sage we may need some coffee in between, and there is no problem if we have to buy it first next corner.)

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