#11474: Elliptic curves should be unique parent structures
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Reporter: SimonKing | Owner: cremona
Type: defect | Status: needs_info
Priority: major | Milestone: sage-6.2
Component: elliptic curves | Resolution:
Keywords: unique parent | Merged in:
Authors: Simon King | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Changes (by pbruin):
* cc: cremona (added)
Comment:
I think this would still be good to fix. The consensus seems to be that
an `EllipticCurve` object should be defined uniquely by its base ring and
the coefficients ''a'',,''i'',,, and nothing else. The datum consisting
of the base ring and the Weierstrass coefficients is "enough" in the sense
that if these are identical for two given elliptic curves, then there is a
canonical isomorphism between them.
However, in the `sage-nt` discussion linked to above, there was some
discussion about situations like the following. The user creates an
elliptic curve ''E'' (say over '''Q''') without using Cremona's database,
and computes generators for its Mordell-Weil group. Now he later tries to
load the same curve from the database, but the generators don't agree.
Should the two curves be identical or not?
Intuitively, I would say they ''should'' be identical; the fact that there
is no canonical basis for the MW group does not justify breaking the
unique parents convention (equality of parents, as determined by some
defining data, implies identity).
On the question of whether the generators from the database should
override the ones computed by the user, I think the answer is no. The
database provides a convenient way to avoid a possibly long computation,
but the basis returned by it has no mathematical property that makes it
preferred over any other one.
Compare to `NumberField`: it seems strange to call two number fields non-
identical if they agree in all respects except that different bases for
the unit group have been computed. (Actually, `NumberField` may be a bad
example because the current implementation also uses various other
parameters to decide whether to construct a new instance. The point is
that it does ''not'' take any basis for the unit group into account.)
--
Ticket URL: <http://trac.sagemath.org/ticket/11474#comment:7>
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