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Nice scripts and programs for PARI/GP
Website

The Rosetta Code
includes a large number of examples of GP scripts.
Series, products, integrals
Elliptic curves
 [get there] descent via isogeny on elliptic curves which have a rational
torsion point of order N (by V. Flynn and C. Grattoni).
 [get it] torsion over number fields, isogeny graphs, divisors, Vélu
formulae (by François Brunault).
 [get it]. 2descent over number fields. Works quite well for fields of
degree less than 3, and is well suited for handling several quadratic twists
of a common curve. It gives an alternative to John Cremona's mwrank even over
Q. Of independant interest, the submodule
qfsolve solves efficiently any ternary quadratic equation, even
not in diagonal (Legendre) form (by Denis Simon).

[get it].
Compute Heegner points, padic or complex (by Peter Green).
 [get it].
Derived periods and the MazurTate circle pairing (by
Henri Darmon and Massimo Bertolini).
 [get original] / [get new version]. Hecke
operators on S_{2}(Γ_{0}(N)),
by
Joe Wetherell, William Stein, Dominique Bernardi, Bernadette PerrinRiou and
Karim Belabas.
 [get it]. Various scripts by
John Cremona (Heegner points, division polynomials, Weil pairing,
group structure over ℤ/pℤ...)
 [get there] The APIP library (GPL) for computing standard and optimized
variants of most cryptographic pairing, by Jérome Milan

[get it] (OBSOLETE). The 'ellsea' package, an implementation of the
SchoofElkiesAtkin algorithm over a prime field,
by Christophe Doche and Sylvain Duquesne. The functionality
is now available natively in pari2.5, using the
seadata official package.

[get it] (OBSOLETE). An implementation of Satoh's pointcounting algorithm
(by KimEe Yeoh).
The functionality is now available natively in pari2.6:
ellcard .

[get it] (OBSOLETE).
Compute heuristically the analytic rank of an elliptic curve (by Tom Womack).
The functionality is now available natively in pari2.5:
ellanalyticrank .
Miscellanea

[get it] Ramanujan's τ function (naive O(sqrt(p)) algorithm,
by Henri Cohen).

[get there]
A Database of Local Fields, by John W. Jones and David P. Roberts.

[get there]
Jeroen Demeyer's
fullfactor package, implementing a large set of
integer factorization helpers for integers of a special form (includes tables
from the Cunningham project).

[get there]
A page on diagonal quartic surfaces, containing GP scripts and PARI
programs (Picard groups and point counting); by Martin Bright.

[get there]. Large library of GP routines, developed by
Fernando RodriguezVillegas and
Ariel Pacetti: quaternion algebras, binary quadratic forms, conics,
jacobian of cubics/quartics, Igusa invariants, Dedekind's eta function on CM
points, skewsymetric matrices, elliptic units, elliptic dilogarithm, padic
gamma function, Brandt matrices, lifting matrices, polynomials in several
variables, vector handling routines, various factoring algorithms over Z
(p1, Lenstra), Ktheory of number fields and zeta values
(LichtenbaumZagier conjectures), statistics...
 [get there] Miscellaneous scripts
written by Jörg Arndt (continued
fractions, Pell equation, recurrent sequences, gaussian normal basis,
hypergeometric functions, singular value decomposition)
 [get there]
Miscellaneous scripts written by Max Alekseyev:
number of Hamiltonial paths and cycles in graphs,
inversion of Euler Totient function, binomial coefficients modulo primes
number of subgroups of an abelian group, period of recurrent sequences modulo
primes, empirical recurrent formulas with polynomial coefficients, generation
of integer partitions, number of irreducible multivariate polynomials over
finite fields.

[get
there] Bianchi.gp, by Alexander Rahm: compute fundamental domains for the Bianchi groups.
Books / Lectures

[get there]
A book (Oxford Graduate Text in Math 13, 2007, 232 pages) by
Fernando RodriguezVillegas, containing a huge number of sample
GP programs.

[get there]. A series
of introductory lectures about number theory, with a bias towards explicit
computations, by William Stein. It contains an introduction to GP and many
fun scripts.

[get there].
The SIAM 100Digit Challenge book is a gourmet feast in ten courses on
(high precision) numerical analysis. Some GP scripts are available on the
book web site.
