A006962 Supersingular primes of the elliptic curve X_0 (11).

2, 19, 29, 199, 569, 809, 1289, 1439, 2539, 3319, 3559, 3919, 5519, 9419, 9539, 9929, 11279, 11549, 13229, 14489, 17239, 18149, 18959, 19319, 22279, 24359, 27529, 28789, 32999, 33029, 36559, 42899, 45259, 46219, 49529, 51169, 52999, 55259

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The primes for which A006571(p) == 0 (mod p) are called supersingular for the elliptic curve "11a3" and form sequence A006962. A prime p>2 is in A006962 if and only if A006571(p) = 0. - Michael Somos, Dec 25 2010

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Joerg Arndt, Table of n, a(n) for n = 1..747 (first 60 terms from Seiichi Manyama)
• S. Lang and H. F. Trotter, Frobenius Distribution in GL_2-Extensions Lect Notes Math. 504, 1976, see p. 267.

Crossrefs

Cf. A006571.

Programs

Mathematica

maxPi = 500; QP = QPochhammer; s = q*(QP[q]*QP[q^11])^2 + O[q]^(Prime[ maxPi] + 1); Reap[Do[If[Mod[SeriesCoefficient[s, p], p] == 0, Print[p]; Sow[p]], {p, Prime[Range[maxPi]]}]][[2, 1]] (* Jean-François Alcover, Nov 29 2015, adapted from PARI *)

PARI

forprime(p=2, 2999, if(polcoeff(x * sqr(eta(x + O(x^p)) * eta(x^11 + O(x^p))), p)%p == 0, print1(p","))) /* Michael Somos, Dec 25 2010 */

PARI

\\ gp -s 30G < A006962.gp
{ N = 10^8 + 2;
default(seriesprecision,N);
V = Vec((eta(q) * eta(q^11))^2);
forprime(p=2,N, if( V[p]%p == 0, print1(p,", ") ) );
} \\ Joerg Arndt, Sep 10 2016

Ruby

require 'prime'
def A006962(n)
  ary = []
  cnt = 1
  Prime.each(10 ** 7){|p|
    a = Array.new(p, 0)
    (0..p - 1).each{|i| a[(i * i) % p] += 1}
    s = 0
    (0..p - 1).each{|i|
      s += a[(i * i * i - 4 * i * i + 16) % p]
      break if s > p
    }
    if p == s
      ary << p
      cnt += 1
      return ary if cnt > n
    end
  }
end # Seiichi Manyama, Sep 10 2016

Extensions

a(29)-a(38) from Michael Somos, Dec 25 2010