id,summary,reporter,owner,description,type,status,priority,milestone,component,resolution,keywords,cc,merged,author,reviewer,upstream,work_issues,branch,commit,dependencies,stopgaps
27413,Implement picard group and unit group for non-maximal orders,klui,,"The goal of this ticket is to compute the unit group (as a subgroup of the unit group of the maximal order) and to compute the picard group (these are useful for isomorphism-testing and isogeny-enumerating of modabvars.) So the goal is to implement the algorithms here: https://math.uni-paderborn.de/fileadmin/mathematik/AG-Computeralgebra/Publications-klueners/picard.pdf . These have already been implemented in Magma (by the authors of the paper, I believe) so we should double-check with the results there.
Non-maximal orders are not well-supported in Sage (Sage uses PARI for maximal orders but PARI does not have non-maximal orders, except for quadratic number fields). For example,
{{{
sage: K. = NumberField(x^2-5)
sage: O = K.order(a); O.is_maximal()
False
sage: O.ideal(2)
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call last)
in ()
----> 1 O.ideal(Integer(2))
/home/klui/sage/local/lib/python2.7/site-packages/sage/rings/number_field/order.pyc in ideal(self, *args, **kwds)
232 """"""
233 if not self.is_maximal():
--> 234 raise NotImplementedError(""ideals of non-maximal orders not yet supported."")
235 I = self.number_field().ideal(*args, **kwds)
236 if not I.is_integral():
NotImplementedError: ideals of non-maximal orders not yet supported.
}}}
So this might be hard.
== Progress ==
This current branch can:
* return the underlying free module of a fractional ideal of an order given the generators.
* compute the conductor of a non-maximal order as both an ideal of the maximal order and the non-maximal order.
* intersect fractional ideals of non-maximal orders with non-maximal orders.
* return the prime factors of an integral ideal of a non-maximal order.
== Github ==
I've been using Github for version control. See https://github.com/kevinywlui/sage/tree/order for the lastest development. But I'll push to this ticket every now and then.",enhancement,new,major,sage-wishlist,number fields,,"class group, unit group, non-maximal order, picard group",,,,,N/A,,u/klui/implement_picard_group_and_unit_group_for_non_maximal_orders,3df71d581a10a14a5399b79f666560e1fd2dfc6b,,