Thank you all for the temporary solution to my problem arising from an ambitious effort to understand Table 12.12 in David Cox's book Primes of the form x^2+ny^2. As Prof Cremona has stated the existence of *only* four perfect cubes on the imaginary axis is to be discussed under an appropriate topic head.
Sincerely On Thu, Sep 3, 2020 at 1:16 AM John Cremona <john.crem...@gmail.com> wrote: > > > On Wednesday, September 2, 2020 at 5:00:07 PM UTC+1 kks wrote: > >> Yes, I knew the point regarding >> >> >> ndeed, there are 9 imaginary quadratic extensions of Q for which one >> gets integer j-invariant, one of them >> Q[sqrt(-163)], but as 163 mod 4 = 3, one has to compute its j-invariant as >> ellj((1+sqrt(163)*I)/2) >> getting -262537412640768000 >> << >> However on the boundary of the fundamental domain, my calculation shows >> only j-invariants ( positive) which are perfect cubes as >> Q(sqrt(-1)).....12^3 >> Q(sqrt(-2))......20^3 >> Q(sqrt(-4))......66^3 >> Q(sqrt(-7))......255^3 >> and the above almost integer. Are there any others on the Imaginary axis? >> > > No. cm_j_invariants_and_orders(QQ) gives all 13 imaginry quadratic orders > of class number 1, from which you can recover the 13 associated imaginary > discriminants D. Most of these are congruent to 1 mod 4 so the j-value is > j((1+sqrt(d))/2) , only those which are 0 mod 4 are on the imaginary axis > with values j(sqrt(D)/2) as in your list. > > THere is a big theory of complex multiplcation behind these facts, but I > don'y think that "gp in Sage" is an accurate sub ject > line for discussion about that. > > John Cremona > > > >> >> >> On Mon, Aug 31, 2020 at 2:28 AM Dima Pasechnik <dim...@gmail.com> wrote: >> >>> On Sun, Aug 30, 2020 at 9:24 AM Dima Pasechnik <dim...@gmail.com> wrote: >>> > >>> > On Sun, Aug 30, 2020 at 5:50 AM Surendran Karippadath >>> > <kksin...@gmail.com> wrote: >>> > > I evaluated the j-invariant in Pari/gp In SageMathCell >>> > > ? \p 50 >>> > > ? ellj(sqrt(163.0)*I) >>> > > %1 = 68925893036109279891085639286944512.000000000163739 >>> > >>> > Sage has this function too (it calls Pari, so that's not an >>> > independent confirmation that this number is (not) an integer: >>> > sage: elliptic_j(sqrt(163)*I,prec=500) >>> > >>> 6.89258930361092798910856392869445120000000001637386442092346075751855217523117650690239250072955532985645916831850173541132959651401661828116253839333e34 >>> > >>> > The output in Pari is a bit easier to read: >>> > >>> > ? \p 500 >>> > ? ellj(sqrt(163)*I) >>> > %4 = >>> 68925893036109279891085639286944512.00000000016373864420923460757518552[...] >>> > >>> > Is it one of these "almost integers" (unless it's a bug, and this >>> > number must be an integer, I don't know - number theorists, please >>> > step forward!), such as >>> > ? \p 500 >>> > ? exp(sqrt(163)*Pi) >>> > %3 = >>> 262537412640768743.99999999999925007259719818568887935385633733699086270[...] >>> > >>> > Or a bug in Pari/GP ? >>> >>> If I read the discussion after Cor. 42 in >>> http://people.maths.ox.ac.uk/greenbj/papers/ramanujanconstant.pdf >>> right, this is not an integer. >>> Indeed, there are 9 imaginary quadratic extensions of Q for which one >>> gets integer j-invariant, one of them >>> Q[sqrt(-163)], but as 163 mod 4 = 3, one has to compute its j-invariant >>> as >>> ellj((1+sqrt(163)*I)/2) >>> getting -262537412640768000 >>> >>> >>> > >>> > >>> > > >>> > > Furthermore the Cube-root of the j-invariant I obtained >>> > > ? (ellj(sqrt(163.0)*I))^(1/3) >>> > > %2 = 410009702400.00077461269365317226812447191214259043 >>> > >>> > the closest integer to (ellj(sqrt(163.0)*I))^(1/3) is 410009702400, >>> > and so you can check >>> > that its cube is quite far from ellj(sqrt(163)*I) >>> > >>> > >>> > >>> > > >>> > > Is it possible to check in Sage with High Precision if the values >>> are Integers. >>> > > >>> > > Thanking you in advance >>> > > >>> > > -- >>> > > You received this message because you are subscribed to the Google >>> Groups "sage-support" group. >>> > > To unsubscribe from this group and stop receiving emails from it, >>> send an email to sage-support...@googlegroups.com. >>> > > To view this discussion on the web visit >>> https://groups.google.com/d/msgid/sage-support/CAGp5ChV21gzwieMCGvywDMxETh3ZUOuOpHaSGUbPNhNHvbVOpg%40mail.gmail.com >>> . >>> >>> -- >>> You received this message because you are subscribed to the Google >>> Groups "sage-support" group. >>> To unsubscribe from this group and stop receiving emails from it, send >>> an email to sage-support...@googlegroups.com. >>> >> To view this discussion on the web visit >>> https://groups.google.com/d/msgid/sage-support/CAAWYfq0kox%3DLYaemNbNdXcDNHKDMxwFaYq9VKASHDoaa1bjZ%3Dg%40mail.gmail.com >>> . >>> >> -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-support+unsubscr...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sage-support/ebafb877-a4e0-4d91-aa0f-85c030de2d7an%40googlegroups.com > <https://groups.google.com/d/msgid/sage-support/ebafb877-a4e0-4d91-aa0f-85c030de2d7an%40googlegroups.com?utm_medium=email&utm_source=footer> > . > -- You received this message because you are subscribed to the Google Groups "sage-support" group. 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