Tables of solutions and other information concerning
Diophantine equations [equations where the variables are constrained
to be integers or rational numbers]:
-
Trinomials
with unusual Galois groups
(x5-5x2-3,
x5-5x2+15,
x7-7x+3,
x8+324x+567, etc.)
-
(Supporting computational data for Nils Bruin's theorem
here)
-
Elliptic
curves of large rank and small conductor
(arXiv preprint;
joint work with Mark Watkins;
to appear in the proceedings of ANTS-VI (2004)):
Elliptic curves over Q of given rank r
up to 11 of minimal conductor or discriminant known;
these are new records for each r in [6,11].
We describe the search method tabulate the top 5 (bottom 5?)
such curves we found for r in [5,11] for low conductor,
and for r in [5,10] for low discriminant.
-
Data and results concerning the elliptic curves
ny2=x3-x arising in the
“congruent number” problem:
-
Transparencies for ANTS-V lecture, in
PS and
PDF
(added 7/2002)
-
All
|n|=8k+7<106 for which the curve
has analytic rank at least 3, listed and used;
-
The same list for
|n|=8k+5<106,
under an additional conjecture;
-
Ditto for
|n|=8k+6<106.
Combining these three, we conclude that every “congruent number”
curve of sign -1 with |n|<106 has positive rank.
-
More rank-3 twists in the same congruence classes: all
|n|=8k+7<107
such that the curve has rank at least 3
assuming the conjecture of Birch and Swinnerton-Dyer; all
|n|=8k+5<107
such that the curve has rank at least 3
under the assumptions of BSD and generalized Gross-Zagier;
and similarly for
|n|=8k+6<107,
where curiously there are significantly more such |n| with k even
than there are with k odd, and the combined list is much longer
than either the 8k+5 or 8k+7 list.
-
Elliptic
curves x3 + y3 = k of high rank
(arXiv preprint;
joint work with Nicholas Rogers, based on his doctoral thesis;
to appear in the proceedings of ANTS-VI (2004)):
The first examples of elliptic curves
Ek: x3 + y3 = k
of ranks 8, 9, 10, 11 over Q,
and thus also the first examples of elliptic curves
xy(x+y)=k with a rational 3-torsion point and rank as high as 11
(the previous record was 8).
We also discuss the problem of finding the minimal curve Ek
of a given rank, in the sense of both |k| and the conductor
of Ek, and we give some new results in this direction.
-
Rational points on the twisted Fermat cubic curves
x3 + y3 = pf z3
(f=1,
f=2)
for certain primes p
(updated 11/2002, to report on
directly-computed values of L'(E,1) and on new experimental data for
p=9k+8
with f=1; and again 4/2003 and 9/2004, to include (p,f)=(9k+8,2) data)
-
Orbit representatives of the 218044170240
minimal
vectors in the rank-128 Mordell-Weil lattice of
y2+y=x3+t65+a6
in characteristic 2 (for the context, see this
preprint, now published in the journal
Experimental Mathematics
[10 (2001) #3, 467-473]).
-
Data concerning
Hall's
conjecture:
small nonzero values of |x3-y2|
-
update 6/2003: Calvo, Sáez and Herranz
propose another search algorithm, which has been used to recover
the all the solutions in my table and a dozen new ones; see
this page for the list and some information
about their new algorithm. I do not know how the two algorithms
are related; I suspect, but have not proved, that they turn out
to be quite close, but that the Calvo-Sáez-Herranz approach
finds more examples by searching in families likely to contain them,
with the downside that one cannot guarantee that it will find
all solutions up to some bound on x,y.
-
My 1988 construction of elliptic curves
y2=x3+ax+b with a
large
integral point (added 6/2003)
-
Fermat
“near-misses”: small values of
|xn + yn - zn|
(n=4,5,...,20; for n=3 see
Dan Bernstein's tables
[added 6/2002: and also
H.Mishima's page on the
n=x3+y3+z3 problem])
-
Elliptic curves over Q with nontorsion
rational
points of low canonical height (added 6/2002)
-
For some results and questions on the corresponding question for
curves over P1(C) etc.,
see my lecture transparencies in
PS
or PDF;
the linear combinations of second Bernoulli functions that I used
are plotted for x in [0,1/2] (a fundamental domain for the infinite
dihedral symmetry group) in
PS and
GIF
for sigma=6, and again for sigma=4
(PS,
GIF
-- for some reason I can't convert these .ps plots to .pdf).
-
An elliptic curve y2=x3-Dx of
rank 13
(updated 5/2002: also rank 14)
- Examples of simple
genus-2
Jacobians with rational torsion points of high order,
and a few other new amusing examples of curves of genus 2
over Q
(updated 6/2002: more rank-1 cases)
- Explicit Hecke correspondences for use in the construction
of towers of modular curves:
-
Explicit complete parametrization of the
cubic
Fermat surface
w3+x3+y3+z3=0
by homogeneous cubics in three variables
(thanks to Dave Rusin for keeping my
original posting
to rec.puzzles and sci.math of these formulas
at his
known-math archives)
Links (see also
this list at the
Number Theory Web):
- Andrew Odlyzko's tables of
zeros of the Riemann zeta function
- Michael Rubinstein's
L-function data
(zeros, special values, etc.)
- Number fields of low degree and
small discriminant (the Bordeaux tables,
initiated by Henri Cohen) or
few ramified primes (John Jones and David Roberts),
and related information; and of
given Galois group up to degree 15
(Jürgen Klüners and Gunter Malle)
-
Tables of
curves of given genus over finite fields
with many rational points,
maintained by Gerard van der Geer and Marcel van der Vlugt
-
John Cremona's tables of
elliptic curve data
-
Interactive interface
to the same data (plus quadratic twists), implemented by
Gonzalo Tornaría
(as recently announced by
Fernando Rodriguez Villegas)
-
William Stein's
modular forms database,
containing some of the same information and also
modular forms of other levels and/or characters
-
Annegret Weng's tables of
Class polynomials for Weber functions associated
with CM elliptic curves of fundamental discriminant down to -422500
-
The
Gebel-Petho-Zimmer tables
of the arithmetic of the “Mordell curves”
y2=x3+k for |k|<104.
These give most of the standard invariants (rank, torsion,
analytic Sha, Mordell-Weil generators, and regulator), together
with the list of integral points. In particular these tables
contain all solutions of the Diophantine inequality
0<|x3-y2|<104.
The data are presented as two megabyte-plus text files, one each for
positive and
negative k.
-
Tom Womack's tables of
elliptic curves of rank up to 11
and lowest conductor known, and of
“Mordell curves”
(elliptic curves y2=x3+k) of high rank
-
Andrej Dujella's list of
elliptic curves of record rank
with prescribed torsion and the record ranks for
infinite families of elliptic curves
with prescribed torsion
-
Michael Stoll's list of
genus-2 curves with small odd discriminant
and small coefficients
-
Graham Everest's page on
“elliptic divisibility sequences”
and related matters (data and heuristics on
prime terms in EDS's, and the
“elliptic Lehmer problem”
on small positive heights)
-
Various useful
PARI-GP routines for computational number theory,
by Fernando Rodriguez Villegas
-
N.J.A. Sloane's searchable
On-Line Encyclopedia of Integer Sequences
-- includes, but is far from limited to, sequences relevant
to algorithmic number theory
Algorithmic
number theory? What's that?
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