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.<a> = NumberField(x^2-5)
sage: O = K.order(a); O.is_maximal()
False
sage: O.ideal(2)
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-10-8da0cc5ea6ab> in <module>()
----> 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.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. Last 10 new commits:

​b5aa206	Add comments to conductor
​ae37da6	Add an OrderIdeal class
​ce1a7ca	Use OrderIdeal as base class of NumberFieldIdeal
​8416412	Generalize free_module to work for OrderIdeals
​dda08ba	Add a README which I will remove at the end
​157ab65	Rename from order_ideal to order_fractional_ideal
​2f2a91d	Fix a typo
​80905e6	Allow conductor to be returned as ideal of self
​d2455f2	Update README to remind to ask sage-nt
​9d09ebf	Implement richcmp for ideals

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​c58a523

Implement intersection of orders with ideals

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. Last 10 new commits:

​e487c7e	Rename from order_ideal to order_fractional_ideal
​a061edc	Fix a typo
​ca83717	Allow conductor to be returned as ideal of self
​8e9337e	Update README to remind to ask sage-nt
​b95b503	Implement richcmp for ideals
​1adfd9d	Implement intersection of orders with ideals
​16b453f	Implement a hash on fractional ideals
​eb87920	Fix bug in intersection
​c1b9b26	Fix remaining bug from conductor
​3df71d5	Implemented prime_factors