Tables

Hilbert class field of totally real fields of degree 2, 3 and 4
This table contains a list of the Hilbert class field of non-principal totally real fields of degree 2, 3 and 4 for all discriminants up to a certain bound.

For degree 2, all discriminants up to 30000 are given for a total of 5763 fields. Class numbers range from 2 to 35. For each non-principal quadratic real field \(k\), an irreducible monic integral polynomial defining an extension \(L\) of the rationals is given. The Hilbert class field of \(k\) is then the compositum of \(k\) and \(L\).

For degree 3, all discriminants up to 150000 are given for a total of 999 fields. Class numbers range from 2 to 13. For each non-principal cubic real field \(k\), an irreducible monic integral polynomial defining the Hilbert class field of \(k\) over the rationals is given.

For degree 4, all discriminants up to 600000 are given for a total of 543 fields. Class numbers range from 2 to 4. For each non-principal quartic totally real field \(k\), an irreducible monic integral polynomial defining the Hilbert class field of \(k\) over the rationals is given.

The Hilbert class field have been computed using the PARI/GP system. The method used is Kummer theory when the class number is 2 and Stark units otherwise (see Computing the Hilbert Class Field of Real Quadratic Fields for the quadratic case, and Stark's Conjectures and Hilbert's Twelfth Problem for the general case).

Hilbert class field of real quadratic fields have been computed with the help of Igor Schein.
Table in PARI/GP format gp (306k)    gp.gz (91k)    
Totally real fields with small root discriminant
This table contains a list of totally real fields of small root discrimants. These fields have been constructed following the original idea of Cohen, Diaz y Diaz, and Olivier (see A table of totally complex number fields of small discriminants, Algorithmic Number Theory (ANTS-III), Lecture Notes in Comput. Sci. 1423, Springer (1998), 381-391) but with totally real fields. That is: they are abelian extensions of totally real fields of small degree.

The table contains totally real number fields of degree 4 to 48, and root discriminant less than 10% above Odlyzko's bounds. The fields are sorted by degree. For each field \(K\) the following pieces of data are given: the difference (as a percentage) with the corresponding Odlyzko bound, an irreducible monic integral polynomial defining \(K\) over the rationals, the factorized discriminant of \(K\), a totally real base field \(k\) such that \(K/k\) is abelian, the conductor and the congruence group of \(K/k\).

All the computations have been done using the PARI/GP system. The polynomials defining these fields have been computed using Stark units (see Stark's Conjectures and Hilbert's Twelfth Problem).
Table in text format txt (15k)    
Totally real quintic dihedral fields
This table contains a list of all totally real fields \(K\) of degree 5 and discriminant up to \(4 \cdot 10^{10}\) whose Galois closure \(N\) has for Galois group the dihedral group of cardinality 10. These fields have been constructed in three steps. First, find suitable real quadratic fields \(k\). Second, construct the fields \(N\) as cyclic degree 5 extensions of \(k\) using Stark units or Kummer theory. Three, find the fields \(K\) as subfields of the fields \(N\).

The table contains 2416 fields. The first entry is the square-root of the discriminant of the field \(K\), the second the discriminant of the corresponding field \(k\), the third is the conductor of \(N/k\), and finally the last entry gives a polynomial defining \(K\) over \(\mathbb{Q}\). Theses fields were used in the paper Numerical Verification of the Stark-Chinburg Conjecture for Some Icosahedral Representations.

All the computations have been done by Igor Schein using the PARI/GP system.
Table in text format txt (113k)    
Rank one quadratic twists of some elliptic curves
These tables contain numerical data on the quadratic twists by negative discriminants of the first smallest (for the conductor) elliptic curves.

The file for each curve consists of a list of vectors (one by line) containing the following data: 1) a fundamental negative discriminant d satisfying additional conditions so that the sign of the functional equation is -1, 2) the short ellinit of the minimal Weierstrass model of the quadratic twist \(E_d\) by d, 3) the product of the Tamagawa numbers of \(E_d\), 4) a generator of the Mordell-Weil group of \(E_d\), 5) the analytic order of the Tate-Shafarevich group of \(E_d\), 6) the index of the subgroup generated by torsion and the Heegner point used in the computation. Note that if the analytic rank of \(E_d\) is greater than 1 then the last three entries are zero.
The lines are ordered by decreasing discriminant and contains all the fundamental discriminants satisfying the additional conditions of absolute value less than \(B\). This file can be imported into PARI/GP using the readvec function.

This is a joint work with C. Delaunay, see Regulators of rank one quadratic twists for more details. All the computations have been done using the PARI/GP system.
Curve \(11a1\) with \(B = 1 600 000\) (\(222 900\) curves) gp.gz (118Mb)    
Curve \(14a1\) with \(B = 1 600 000\) (\(70 944\) curves) gp.gz (46Mb)    
Curve \(15a1\) with \(B = 1 600 000\) (\(76 004\) curves) gp.gz (53Mb)    
Curve \(17a1\) with \(B = 1 600 000\) (\(229 663\) curves) gp.gz (159Mb)    
Curve \(19a1\) with \(B = 1 600 000\) (\(231 031\) curves) gp.gz (123Mb)    
Rubin-Stark elements over quadratic fields
These tables contain the Rubin-Stark elements that were computed for the paper Testing the Congruence Conjecture for Rubin-Stark Elements with D. Solomon. Each file corresponds to a class of examples of the paper and contains a vector with one component for each example in the class. In turn, each component contains the following data: the discriminant \(d_k\) of a real quadratic \(k\), a conductor \(f\) of \(k\) (given by its HNF PARI/GP representation), the irreducible polynomial over \(\mathbb{Q}\) of a generating element \(\lambda\) of a totally real abelian extension \(K^+\) of \(k\) with conductor \(f\), the prime \(p\), an integer \(a\) and two units (or \(p\)-units) \(u_1\) and \(u_2\), expressed as polynomials in \(\lambda\), such that \(\frac{1}{a} \otimes (u_1 \wedge u_2)\) is the Rubin-Stark element associated to \(K^+/k\), \(d=2\) and the set \(S^1\) of the infinite places of \(k\), the primes ramified in \(K/k\) and the primes in \(k\) above \(p\).

All the computations have been done using the PARI/GP system.
Examples B1 to B12 gp (11k)    
Examples C1 to C16 gp (3k)    
Examples D1 to D14 gp (2k)    
Examples E1 to E6 gp (236k)    

X.-F. Roblot Last change: 27 February 2017, 10:03