Re: A question

[John Cremona <John.Cremona@nottingham.ac.uk>]

Those three curves have analytic Sha of order 1, 4, 1 respectively.

The models [0,0,0,A,B] are not global minimal models for the parameters
(61,62),(80,80),(93,94)
which cause you problems, and this could well be the reason.

You may be interested in the scripts written by my student Tom Womack
which compute analytic rank and analytic order of sha for curves whose
conductor is not too large.  See
http://www.maths.nott.ac.uk/personal/pmxtow/BG.gp

In your three cases the output is as follows; the output vector contains
[rank, L^(r)(1), approximate |Sha|].

================================================================================

(09:03) gp > ellanalyticrank(ellinit([0,0,0,61,62]))
INPUT CURVE NOT MINIMAL
MINIMISED TO [1, -1, 0, 4, 0]
Summing 44 a_n terms
Rank is even
L^(0)=1.3738172276808556288946844204737122434
time = 100 ms.
%6 = [0, 1.3738172276808556288946844204737122434,
0.99999957636300480694027822639775915684]

(09:03) gp > ellanalyticrank(ellinit([0,0,0,80,80]))
INPUT CURVE NOT MINIMAL
MINIMISED TO [0, 0, 1, 5, 1]
Summing 186 a_n terms
Rank is even
L^(0)=10.232592771285236568438340493524215426
time = 333 ms.
%8 = [0, 10.232592771285236568438340493524215426,
3.9999991310198928452874527955085657339]

(09:03) gp > ellanalyticrank(ellinit([0,0,0,93,94]))
INPUT CURVE NOT MINIMAL
MINIMISED TO [1, -1, 0, 6, 0]
Summing 19 a_n terms
Rank is even
L^(0)=0.81997852216802397053558909545998838983
time = 50 ms.
%7 = [0, 0.81997852216802397053558909545998838983,
1.0000000794045783520078587165958268501]

================================================================================

John Cremona

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