David is right. In Magma, when you ask for the discriminant of a number field you get the discriminant of its defining poynomial which is a multiple of the field discriminant. You have to ask for the discriminant of its ring of integers:
> R<x> := PolynomialRing(Rationals()); > L := NumberField(x^7 - x^5 - 2*x^4 - 2*x^3 + 2*x^2 - x + 4); > Discriminant(Integers(L)); 30558784 > Factorization(Integers()!Discriminant(Integers(L))); [ <2, 6>, <691, 2> ] On Fri, 28 Sep 2018 at 10:21, David Roe <roed.m...@gmail.com> wrote: > This looks like a discrepancy between the discriminant of the polynomial > and the discriminant of the number field (which should be the discriminant > of the maximal order). Sure enough: > > sage: O = L.maximal_order() > sage: O.discriminant().factor() > 2^6 * 691^2 > sage: O.basis() > [4/7*a^6 + 6/7*a^5 + 5/7*a^4 + 3/7*a^3 + 1/7*a + 1/7, a, a^2, a^3, a^4, > a^5, a^6] > > The power basis is not 7-maximal. You're asking Sage and Magma for > different things: > sage: discriminant(x^7 - x^5 - 2*x^4 - 2*x^3 + 2*x^2 - x + 4).factor() > 2^6 * 7^2 * 691^2 > > I'm fairly confident that if you ask Magma for the discriminant of W, > you'll lose the factor of 7^2. > David > > On Fri, Sep 28, 2018 at 4:54 AM Harry Smit <harryjustuss...@gmail.com> > wrote: > >> The following computations were done on cocalc.com, version'SageMath >> version 8.3, Release Date: 2018-08-03' >> >> There seems to be a disagreement with Sage and Magma over the >> discriminant of the number field defined by adjoining a root of the >> following polynomial to Q: >> >> x^7 - x^5 - 2*x^4 - 2*x^3 + 2*x^2 - x + 4 >> >> In Sage, I used as input: >> >> L.<a> = NumberField(x^7 - x^5 - 2*x^4 - 2*x^3 + 2*x^2 - x + 4) >> L.absolute_discriminant().factor(); >> >> to obtain >> >> 2^6 * 691^2 >> >> In Magma, I used as input >> >> L := NumberField(x^7 - x^5 - 2*x^4 - 2*x^3 + 2*x^2 - x + 4); >> g := DefiningPolynomial(L); >> W := IntegerRing(); >> Factorization(W ! Discriminant(g)) >> >> to obtain >> >> [ <2, 6>, <7, 2>, <691, 2> ] >> >> Checking this by hand, it seems that Magma is correct: the polynomial has >> a double root (mod 7) at 5, hence 7 ramifies in the number field. >> >> The problem is not the factor function, this was only added for clarity. >> >> -- >> You received this message because you are subscribed to the Google Groups >> "sage-devel" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to sage-devel+unsubscr...@googlegroups.com. >> To post to this group, send email to sage-devel@googlegroups.com. >> Visit this group at https://groups.google.com/group/sage-devel. >> For more options, visit https://groups.google.com/d/optout. >> > -- > You received this message because you are subscribed to the Google Groups > "sage-devel" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-devel+unsubscr...@googlegroups.com. > To post to this group, send email to sage-devel@googlegroups.com. > Visit this group at https://groups.google.com/group/sage-devel. > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
