Improvements in the computation of ideal class groups of imaginary quadratic number fields
نویسنده
چکیده
We investigate improvements to the algorithm for the computation of ideal class groups described by Jacobson in the imaginary quadratic case. These improvements rely on the large prime strategy and a new method for performing the linear algebra phase. We achieve a significant speed-up and are able to compute ideal class groups with discriminants of 110 decimal digits in less than a week.
منابع مشابه
Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields with Class Numbers 7 and 11
Motivated by a constructive realization of dihedral groups of prime degree as Galois group over the field of rational numbers, we give an explicit construction of the Hilbert class fields of some imaginary quadratic fields with class numbers 7 and 11. This was done by explicitly evaluating the elliptic modular j -invariant at each representative of the ideal class of an imaginary quadratic fiel...
متن کاملSubexponential Class Group Computation in Quadratic Orders (abstract)
In 1989, the first subexponential algorithm for computing the class group of an imaginary quadratic order was introduced by Hafner and McCurley. Their algorithm is based on an integer factorization algorithm due to Seysen, and is conditional on the truth of the Extended Riemann Hypothesis. Not long after, their result was generalized to arbitrary algebraic number fields by Buchmann. Efficient v...
متن کاملEuclidean Ideals in Quadratic Imaginary Fields
— We classify all quadratic imaginary number fields that have a Euclidean ideal class. There are seven of them, they are of class number at most two, and in each case the unique class that generates the class-group is moreover norm-Euclidean.
متن کاملEuclidean Ideals in Quadratic
— We classify all quadratic imaginary number fields that have a Euclidean ideal class. There are seven of them, they are of class number at most two, and in each case the unique class that generates the class-group is moreover norm-Euclidean.
متن کاملQuadratic Fields with Special Class Groups
For every prime number p > 5 it is shown that, under certain hypotheses on x e Q , the imaginary quadratic fields Q( \/x2p 6xf + 1 ) have ideal class groups with noncyclic p-parts. Several numerical examples with p = 5 and 7 are presented. These include the field Q(v/-4805446123032518648268510536). The 7-part of its class group is isomorphic to C(7) x C(7) x C(7), where C(n) denotes a cyclic gr...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Adv. in Math. of Comm.
دوره 4 شماره
صفحات -
تاریخ انتشار 2010