On Sun, Aug 30, 2020 at 9:24 AM Dima Pasechnik <dimp...@gmail.com> wrote: > > On Sun, Aug 30, 2020 at 5:50 AM Surendran Karippadath > <kksinfin...@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+unsubscr...@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+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/CAAWYfq0kox%3DLYaemNbNdXcDNHKDMxwFaYq9VKASHDoaa1bjZ%3Dg%40mail.gmail.com.
