# Department of Mathematics and Statistics

## Dan Yasaki

### 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.

### 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. $n_{29}$