As John Cremona said at the sage.forum: "But why would Sage be
computing the class group in order to factor 2 in K?"
For me that is strange too. Since it would be easier to compute the
factorization of an ideal generate by a prime.
For example: Using the Proposition that asserts that if K is a number
field and the ring of integers is of the form O_K=Z[alpha] (where f(t)
is the minimal polynomial over Z of alpha) then to compute the
factorization of the ideal <p>O_K where p is a rational prime is
enough to factorize f(t)mod p. That is, if
f(t)=f_1(t)^r_1...f_s(t)^r_s (mod p) then the factorization on prime
ideal is of the form
<p>O_K=<p,f_1(alpha)>^r_1...<p,f_s(alpha)>^r_s.
Enrique
On 29 dic, 21:42, Bill Hart <[EMAIL PROTECTED]> wrote:
> It presumably can't compute the class group, because of the proof=true
> thing. Basically if you want a proven result, it is going to take
> forever and will need more primes than Pari has precomputed.
>
> It's not clear to me if SAGE is actually catching the error message,
> or if it is just raising an exception because Pari did something it
> wasn't expecting (i.e. print an error message).
>
> Bill.
>
> On 29 Dec, 20:27, mabshoff <[EMAIL PROTECTED]
>
> dortmund.de> wrote:
> > On Dec 29, 9:17 pm, [EMAIL PROTECTED] wrote:
>
> > > Hello:
>
> > > I am working at my Number Theory lectures and I have found a bug (?).
> > > This is
> > > the output:
>
> > > /////////////////// SAGE 2.9.1 ///////////////////
> > > sage: K.<a>=CyclotomicField(23)
> > > sage: O=K.maximal_order()
> > > sage: (2*O).factor()
> > > *** Warning: large Minkowski bound: certification will be VERY long.
> > > Traceback (most recent call last):
> > > File "<stdin>", line 1, in <module>
> > > File "/home/notebook/sage_notebook/worksheets/admin/3/code/13.py",
> > > line 4, in <module>
> > > exec compile(ur'(Integer(2)*O).factor()' + '\n', '', 'single')
> > > File
> > > "/usr/local/sage/local/lib/python2.5/site-packages/sympy/plotting/",
> > > line 1, in <module>
>
> > > File "sage_object.pyx", line 92, in
> > > sage.structure.sage_object.SageObject.__repr__
> > > File
> > > "/usr/local/sage/local/lib/python2.5/site-packages/sage/structure/factor\
> > > ization.py", line 187, in _repr_
> > > t = str(self[i][0])
> > > File
> > > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > > ld/number_field_ideal.py", line 218, in __repr__
> > > return "Fractional ideal %s"%self._repr_short()
> > > File
> > > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > > ld/number_field_ideal.py", line 235, in _repr_short
> > > return '(%s)'%(', '.join([str(x) for x in self.gens_reduced()]))
> > > File
> > > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > > ld/number_field_ideal.py", line 553, in gens_reduced
> > > dummy = self.is_principal(proof)
> > > File
> > > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > > ld/number_field_ideal.py", line 714, in is_principal
> > > bnf = self.number_field().pari_bnf(proof)
> > > File
> > > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > > ld/number_field.py", line 1464, in pari_bnf
> > > self.pari_bnf_certify()
> > > File
> > > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > > ld/number_field.py", line 1497, in pari_bnf_certify
> > > if self.pari_bnf(certify=False, units=True).bnfcertify() != 1:
> > > File "gen.pyx", line 6474, in sage.libs.pari.gen._pari_trap
> > > sage.libs.pari.gen.PariError: not enough precomputed primes, need
> > > primelimit ~ (35)
>
> > Hi Enrique,
>
> > this looks like a bug to me. I have seen this issue discussed before,
> > but I cannot find any ticket in our bug tracker that relates to it. So
> > I am hoping for somebody more familiar with the pari interface to
> > voice an opinion.
>
> > Cheers.
>
> > Michael
>
> > > But if you type the following lines using gp interface, it works:
>
> > > sage: K=gp.bnfinit(cyclotomic_polynomial(23))
> > > sage: gp.idealfactor(K,2)
>
> > > [[2, [1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
> > > 0]~, 1, 11, [1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
> > > 0, 0]~], 1; [2, [1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0,
> > > 0, 0, 0, 0]~, 1, 11, [1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0,
> > > 0, 0, 0, 0, 0]~], 1]
>
> > > All the best,
>
> > > Enrique
>
> > > --------------------------------------------------------------------------
> > > Mensaje enviado mediante una herramienta Webmail integrada en *El Rincon*:
> > > ------------->>>>>>>> https://rincon.uam.es <<<<<<<<--------------
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