On Wednesday, March 21, 2018 at 1:18:41 PM UTC, Dima Pasechnik wrote: > > > > On Wednesday, March 21, 2018 at 9:23:26 AM UTC, John Cremona wrote: >> >> >> >> On 20 March 2018 at 23:57, Dima Pasechnik <[email protected]> wrote: >> >>> >>> >>> On Tuesday, March 20, 2018 at 4:06:46 PM UTC, John Cremona wrote: >>>> >>>> Working with your degree 8 polynomial over Q is almost certainly >>>> better. I would also recommend reducing the defining polynomial first: >>>> >>>> sage: R.<g> = QQ[] >>>> sage: pol = g^8 - 5661818/709635*g^7 + 11951452814641/503581833225*g^6 >>>> - 5464287298588/167860611075*g^5 + 42311165180509/503581833225*g^4 + >>>> 290446480816/167860611075*g^3 + 6817133713732/503581833225*g^2 - >>>> 11294971392/55953537025*g + 2238425344/503581833225 >>>> sage: K.<a> = NumberField(pol) >>>> sage: K1=K.optimized_representation()[0]; K1 >>>> Number Field in a1 with defining polynomial x^8 - 2*x^7 - >>>> 2073127276349*x^6 - 585042438455127612*x^5 + >>>> 17251120619520968221641540*x^4 >>>> + 47323235466058260399591984538122*x^3 + >>>> 52569579991119152255555179191805210311*x^2 + >>>> 26979907667586120684167115024265757878264932*x + >>>> 5304889912416030130201287805372669997413025784321 >>>> >>>> -- not obviously a lot better, but at least it has integer >>>> coefficients. We can easily find its Galois group abstractly: >>>> >>> >>> Thanks---I was not aware about optimized_representation(). >>> >> >> Underneath it's the pari function "polred" which does the work. It's a >> good way of getting a nicer polynomial defining the same number field. In >> most cases, calling it with two isomorphic fields will return the same nice >> version; not always, so there is the version polredabs which is guaranteed >> to always give the same output for any polynomial defining the field. (We >> use this in the LMFDB.) >> >> >>> >>> Let me explain a bit what I'm trying to do. >>> >>> Given 7 points in P^2 with coefficients in Q, I need >>> 1) to find an irreducible cubic q through them s.t. q intersects qbar >>> (it's complex conjugate) in 9 distinct real points, and >>> >> >> Assuming that your 7 points are in general position (no three on a line, >> no 6 on a conic), the cubics through them form a 2-dimensional family (they >> are all linear combinations of 3 of them, up to scaling). One way to get >> your q would be to take two such rational cubics q1 and q2 which intersect >> in 9 points and then let q=q1+i*q2, which intersects qbar in the same 9 >> points. >> > > Thanks for pointing out an option of rational cubics. It might be easier > to work with them, right? > Is there a way to do a rational parametrisation of cubcs over Q[i] in Sage? >
Sage calls Singular's https://www.singular.uni-kl.de/Manual/4-0-3/sing_1402.htm For some reason I fail to comprehend, the latter only does the job over Q. It seems that the straightforward computation of resultants (w.r.t. to x, y and z) of the equation of the (rational) cubic (assuming its singular point p has x nonzero) and the equation of the line joining p with the point (0:1:w), with w a parameter will return a polynomial with a linear factor in x, y, resp. z, depending upon powers of w, thus providing the expressions, valid over any field (surely any char. 0 field). -- You received this message because you are subscribed to the Google Groups "sage-nt" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/sage-nt. For more options, visit https://groups.google.com/d/optout.
