OFFSET
1,2
COMMENTS
If we consider the formulas for the Eisenstein series at elliptic integral singular values: E_4(q) = (2*K(k)/Pi)^4*(1-k^2*(1-k^2)) and E_6(q) = (2*K(k)/Pi)^6*(1-2*k^2)*(1+(1/2)*k^2*(1-k^2)), where K is the elliptic integral of the first kind and k is the elliptic modulus, then we may insert this into eta(q)^24 = (E_4(q)^3 - E_6(q)^2)/1728. This leads to the result that eta(sqrt(-n))/sqrt(K(k)/Pi) = (4*k^2*(1-k^2))^(1/12) = (1/G_n)^2, where k = sqrt(lambda(sqrt(-n))) and G_n is Ramanujan's G-function (class invariant). Note that this sequence uses the square free part of n as elliptic modulus and so the result differs in some cases by some factor. We find the minimal polynomials with roots in the algebraic number k in A389848.
If n is from {3, 11, 19, 43, 67, 163} then a(n) = 6 and n == 3 (mod 8), and a(7) = 2 and 7 == 7 (mod 8). These are prime discriminants of imaginary quadratic fields with class number 1.
If n is from {5, 13, 37} then a(n) = 4. These are prime n such that Q(sqrt(-n)) has class number 2.
if n is from {23, 31} then a(n) = 6 and n == 7 (mod 8), if n is from {59, 83, 107, 139, 211, 283, 307, 331, 379, 499, 547, 643, 883, 907} then a(n) = 18 and n == 3 (mod 8). These are prime discriminants of imaginary quadratic fields with class number 3.
if n is from {47, 79, 103, 127} then a(n) = 10 and n == 7 (mod 8), if n is from {131, 179, 227, 347, 443, 523, 571, 619, 683, 691, 739, 787, 947, 1051, 1123, 1723, 1747, 1867, 2203, 2347, 2683} then a(n) = 30 and n == 3 (mod 8). These are prime discriminants of imaginary quadratic fields with class number 5.
LINKS
Mathematics Stack Exchange, Additional values of Dedekinds eta function in radical form (2019).
Mathematics Stack Exchange, What is the exact value of eta(6i)? (2015).
Eric Weisstein's World of Mathematics, Elliptic Integral Singular Value.
EXAMPLE
The minimal polynomials with a real root in eta(sqrt(-n))/sqrt(K(sqrt(lambda(sqrt(-A007913(n)))))/Pi) are:
n
1 x - 1
2 x^8 + 4*x^4 - 4
3 2*x^6 - 1
4 8*x^8 - 1
5 x^4 + x^2 - 1
6 x^48 + 75488*x^36 + 108672*x^24 - 145408*x^12 + 4096
7 2*x^2 - 1
8 4*x^16 + 28*x^8 - 1
9 19683*x^24 + 486*x^12 - 1
10 x^16 + 188*x^12 - 12*x^8 - 112*x^4 + 16
11 2*x^6 - 4*x^4 + 4*x^2 - 1
12 268435456*x^48 - 851968*x^24 + 1
13 x^4 + 3*x^2 - 1
14 x^32 + 632*x^28 - 520*x^24 + 6528*x^20 - 13776*x^16 + 8704*x^12 + 3968*x^8 - 3584*x^4 + 256
15 64*x^12 - 56*x^6 + 1
16 8192*x^16 + 512*x^8 - 1
17 x^4 - x^3 - 2*x^2 - x + 1
18 282429536481*x^48 + 433910947680*x^36 + 3333197952*x^24 - 52254720*x^12 + 4096
19 2*x^6 + 4*x^4 + 4*x^2 - 1
20 4096*x^32 - 3072*x^24 + 8128*x^16 - 752*x^8
21 x^24 + 168*x^18 + 142*x^12 - 168*x^6 + 1
22 x^16 + 4616*x^12 - 120*x^8 - 544*x^4 + 16
23 8*x^6 + 12*x^4 + 4*x^2 - 1
.
Some expressions for K(sqrt(lambda(sqrt(-r)))) are:
r
1 (1/4)*Gamma(1/4)/sqrt(Pi)
2 (1/2^(13/4))*sqrt(sqrt(2)+1)*Gamma(1/8)*Gamma(3/8)/sqrt(Pi)
3 (1/2^(7/3))*3^(1/4)*Gamma(1/3)^3/sqrt(Pi)
... Further values are found in the World of Mathematics link.
PROG
(PARI)
default(realprecision, 500);
lambda(n) = (sqrt(2)*eta(n/2, 1)*eta(2*n, 1)^2/eta(n, 1)^3)^8;
polynomial(n) = my(po = factor(numerator(subst(polclass(-(n*4)), 'x, 256*(1-x^12/4)^3/(x^12/4)^2)))[, 1], l = lambda(I*sqrt(n)), et = (4*l*(1-l))^(1/12)); for(k=1, #po, if(subst(po[k], 'x, et)<10^(-10), return(po[k])));
a(n) = if(core(n)==n, #Vec(polynomial(n))-1, print("This function suports only square-free n."));
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Thomas Scheuerle, Oct 20 2025
STATUS
approved