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.
