Imaginary quadratic torsion data

First batch

First Batch Data for free part and torsion part of the Voronoi homology of \(\textrm{GL}_2( \mathbb{Z}[\sqrt{-1}])\).

We should eventually have the data for all levels up to norm 60,000. That includes 23,765 levels. The columns are as follows. (Update:The size of torsion gets large enough that factoring is taking a non-trivial amount of time. The unfactored data is available upon request.)

  1. ID: This is the ID of the level from master list of levels. In particular, the level is \(\mathfrak{n} = (LevelList[ID])\).
  2. Norm of \(\mathfrak{n}\).
  3. Index \([\mathrm{GL}_2(\mathbb{Z}[\sqrt{-1}]):\Gamma_0(\mathfrak{n})]\). (I just compute the order of \(\mathbb{P}^1(\mathcal{O}_F/\mathfrak{n}\)).)
  4. Rank of free part of cohomology. This includes Eisenstein.
  5. Size of torsion \(T = T_1 T_2 T_3\).
  6. Part of torsion \(T_1\) coming from primes that are 1 mod 4.
  7. Part of torsion \(T_2\) coming from prime 2.
  8. Part of torsion \(T_3\) coming from primes that are 3 mod 4.

Pictures: (gnuplot script. Download First Batch Data above first.)

p-parts

\(p\)-part data: Let \(T\) denote the size of torsion \(T = \prod p^{m_p}\), \(p\) prime. Let \(a_p = p^{m_p}\), and let \(r_p\) denote the rank of the \(p\)-primary part of the torsion.
  1. ID
  2. norm
  3. index
  4. \(T\)
  5. \(a_2\)
  6. \(a_3\)
  7. \(a_5\)
  8. \(a_7\)
  9. \(a_{11}\)
  10. \(a_{13}\)
  11. \(a_{17}\)
  12. \(a_{19}\)
  13. \(a_{23}\)
  14. \(a_{29}\)
  15. \(r_{2}\)
  16. \(r_{3}\)
  17. \(r_{5}\)
  18. \(r_{7}\)
  19. \(r_{11}\)
  20. \(r_{13}\)
  21. \(r_{17}\)
  22. \(r_{19}\)
  23. \(r_{23}\)
  24. \(r_{29}\)
\(X_k\) data: Let \(n_p\) denote the number of levels with \(p\)-rank \(k\). xk picture, log(xk) picture, gnuplot script
  1. \(k\)
  2. \(n_{2}\)
  3. \(n_{3}\)
  4. \(n_{5}\)
  5. \(n_{7}\)
  6. \(n_{11}\)
  7. \(n_{13}\)
  8. \(n_{17}\)
  9. \(n_{19}\)
  10. \(n_{23}\)
  11. \(n_{29}\)