#11220: implement listing j-invariants of CM curves over other fields and fix
incorrect remark in the documentation
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Reporter: was | Owner: cremona
Type: enhancement | Status: new
Priority: minor | Milestone: sage-4.7
Component: elliptic curves | Keywords:
Work_issues: | Upstream: N/A
Reviewer: | Author:
Merged: | Dependencies:
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Comment(by cremona):
The imaginary quadratic orders of class number 2 are the maximal orders in
Q(sqrt(-d)) for d in
[-5,-6,-10,-13,-15,-22,-35,-37,-51,-58,-91,-115,-123,-187,-235,-267,-403,-427]
and the order of index 2 in Q(sqrt(-15)). [Reference: many places
including J E Cremona, Abelian Varieties with Extra Twist, Cusp Forms, and
Elliptic Curves Over Imaginary Quadratic Fields, Journal of the London
Mathematical Society 45 (1992) 402-416.]
Using this list it's easy to write a function for the case of any
quadratic field. [For Q(sqrt(5)) itself it is easy to see that the
discriminant must have the form 5*D where D is a negative prime
discriminant; this gives the same list output as David's script.]
For higher degree one could start with Mark Watkins' determination of all
i.q.fields with h<=100 [See
http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01517-5/S0025-5718-03-01517-5.pdf.]
The number of fields can be as large as 3283 (for h=96) but the table
there gives the maximal discriminant in each case --e.g.427 for h=2 -- so
a script like David's is probably the way to go. That table only deals
with maximal orders. There's still the question of non-maximal orders:
I'll ask Mark if he knows if that has been done (it's just an exercise,
but a lengthy one.)
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11220#comment:1>
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