I actually don't get this focus on "singular values". The ModularLambda function (essentially just inverse τ (K'/K)) is a beautiful quadratic kaleidoscope: In[350]:= RootApproximant[N[ModularLambda[Sqrt[-11]], 69]] Out[350]= Root[ 1 - 2096 #1 + 2864 #1^2 - 5632 #1^3 + 13056 #1^4 - 12288 #1^5 + 4096 #1^6 &, 1] What is K'/K of the other five roots? In[351]:= Table[RootApproximant[ N[EllipticK[1 - #]/EllipticK[#] &@Root[%[[1]], k], 69]], {k, 6}] Out[351]= {Sqrt[11], 1/Sqrt[11], Root[25 - 14 #1^2 + 9 #1^4 &, 4], Root[25 - 14 #1^2 + 9 #1^4 &, 3], Root[9 - 14 #1^2 + 25 #1^4 &, 4], Root[9 - 14 #1^2 + 25 #1^4 &, 3]} I also don't see why Wikipedia skipped #8: In[876]:= ToRadicals[RootApproximant[N[ModularLambda[Sqrt[-8]], 69]]] Out[876]= 113 + 80 Sqrt[2] - 4 Sqrt[2 (799 + 565 Sqrt[2])] In[877]:= RootReduce[%] Out[877]= Root[1 - 452 #1 - 122 #1^2 - 452 #1^3 + #1^4 &, 1] In[878]:= Table[ RootApproximant[ N[EllipticK[1 - #]/EllipticK[#] &@Root[%[[1]], k], 69]], {k, 4}] Out[878]= {2 Sqrt[2], Root[64 + 112 #1^2 + 81 #1^4 &, 4], Root[16 - 8 #1^2 + 9 #1^4 &, 4], Root[16 - 8 #1^2 + 9 #1^4 &, 3]} And this behavior appears to encompass the rationals: In[343]:= RootApproximant[N[ModularLambda[Sqrt[-3/2]], 69]] Out[343]= Root[16 - 32 #1 - 120 #1^2 + 136 #1^3 + #1^4 &, 3] In[344]:= Table[RootApproximant[ N[EllipticK[1 - #]/EllipticK[#] &@Root[%[[1]], k], 69]], {k, 4}] Out[344]= {Root[49 + 60 #1^2 + 36 #1^4 &, 3], Root[25 - 4 #1^2 + 4 #1^4 &, 3], Sqrt[3/2], 1/Sqrt[6]} So 3/2 is "the same as" 6 ! --rwg On 2015-07-15 04:24, rwg wrote:
fxtbook: "See Mathworld." Mathworld gives a bogus negative value. Not in DLMF? Censored from https://en.wikipedia.org/wiki/User:Singular_Modulus functions.wolfram.com gives Out[839]= ( 32 3^(2/3) + 3 (9 + 7 Sqrt[33])^(1/3) - 4 3^(1/3) (9 + 7 Sqrt[33])^( 2/3))/(4 (6 (9 + 7 Sqrt[33])^(1/3) + Sqrt[ 3 (-32 3^(2/3) (9 + 7 Sqrt[33])^(1/3) + 9 (9 + 7 Sqrt[33])^(2/3) + 4 3^(1/3) (9 + 7 Sqrt[33]))]))
In[841]:= (EllipticK[1 - %%]/EllipticK[%%])^2 // N
Out[841]= 11.
Out[842]= 1/2 + Sqrt[11]/12 - (-21 Sqrt[3] + 11 Sqrt[11])^(1/3)/6 - (21 Sqrt[3] + 11 Sqrt[11])^(1/3)/6
In[843]:= (EllipticK[1 - %]/EllipticK[%])^2 // N
Out[843]= 11. --rwg -------- Original Message -------- Subject: [math-fun] the 19th singular value Date: 2011-02-18 01:52 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Reply-To: math-fun <math-fun@mailman.xmission.com>
ETA(%E^-(2*SQRT(19)*%PI)) = ((3*SQRT(57)+1)^(1/3)-8/(3*SQRT(57)+1)^(1/3))^(1/8)*(GAMMA(1/38)*GAMMA(5/38)*GAMMA(9/38)*GAMMA(11/38)*GAMMA(7/19)*GAMMA(17/38)*GAMMA(23/38)*GAMMA(25/38)/(38*GAMMA(13/38)*GAMMA(8/19)))^(1/4)/(2*2^(53/152)*%PI)
where Corey's denester neatly whacks the 8th root: (-(8/(1 + 3*Sqrt[57])^(1/3)) + (1 + 3*Sqrt[57])^(1/3))^(1/8) == -(2^(3/8)/3) + ((-1 + 3*Sqrt[57])^(1/3)*(4*(-5 + Sqrt[57]) + (11 + Sqrt[57])*(-1 + 3*Sqrt[57])^(1/3)))/(96*2^(5/8)) .
It also denests LambdaStar[19], but the result is badly undersimplified. Fullsimplify thuggishly reverts to the original Root object, even when given a ComplexityFunction that enormously penalizes _Root ! But it is possible to guess the general form from the denester mess and determine coefficients with PSLQ: LambdaStar[19]==(((-Sqrt[57] + Sqrt[19] + 7*Sqrt[3] + 11)*(3*Sqrt[57] + 1)^(1/3)/ 16) - ((Sqrt[57] + Sqrt[19] - 7*Sqrt[3] + 11)/(2*(3*Sqrt[57] + 1)^(1/3))) - ((Sqrt[19] - 1)/2))/(3*Sqrt[2])
I was unable to Google whether this denesting is known. Is there a table of singular values more extensive than MathWorld's? --rwg Joerg>Let's not forget Broadhurst's very neat expressions:
Message-ID: <200803140250.m2E2oHi4002315@philter.princeton.idaccr.org> Date: Thu, 13 Mar 2008 22:50:17 -0400 From: David Broadhurst <D.Broadhurst@open.ac.uk> To: NMBRTHRY@LISTSERV.NODAK.EDU Reply-To: David Broadhurst <D.Broadhurst@open.ac.uk> Subject: Singular value for Euler's numerus idoneus N=1365
default(realprecision,1050);
a_1 = 550*sqrt(13) + 318*sqrt(39) + 750*sqrt(7) + 433*sqrt(21); a_2 = 1986 + 1145*sqrt(3) + 208*sqrt(91) + 120*sqrt(273); b = ((a_1-a_2)^2 - 2)/2; k_1365 = sqrt(1/2 - b^2/4*sqrt(4-b^2) - (2-b^2)/4*sqrt(1-b^2)); --------------- The denester also reduced the depth of k_1365, but made a proverbial Very Large Expression, more than Mathematica could further chew. _______________________________________________ MathIsFun