Unconditional Class Group Tabulation of Imaginary Quadratic Fields to $|\Delta| < 2^{40}$
Abstract
We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all $\Delta \not \equiv 1 \pmod{8}.$ The group structure is resolved using the factorization of the group order. The $1 \bmod 8$ case was handled using the methods of \cite{jacobson}, including the batch verification method based on the Eichler-Selberg trace formula to remove dependence on the Extended Riemann Hypothesis. Our new method enabled us to extend the previous bound of $|\Delta| < 2 \cdot 10^{11}$ to $2^{40}$. Statistical data in support of a variety conjectures is presented, along with new examples of class groups with exotic structures.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2015
- DOI:
- 10.48550/arXiv.1502.07953
- arXiv:
- arXiv:1502.07953
- Bibcode:
- 2015arXiv150207953M
- Keywords:
-
- Mathematics - Number Theory
- E-Print:
- 26 pages, 2 figures, to appear in Math. Comp