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?
On Mon, Aug 31, 2020 at 2:28 AM Dima Pasechnik <dimp...@gmail.com> wrote: > 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 > . > -- 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/CAGp5ChW04OyTdsVyP8Pnv_qgdcMVoWSqMtCUPR-d%3D2fwF0_9eA%40mail.gmail.com.
