Descent on Cyclic Covers of the Projective Line

Finite Fields

Deal with David Roe's latest patch bomb.


ABC at home

Models for elliptic curves

Function Fields

The main goal of this project is to get the basic infrastructure for function fields into Sage. If time permits, we will also implement Hess's algorithms. See Hess's papers and talks.

People: William Stein, Maarten Derickx, Peter Bruin, Jan Tuitman, Max Flander, Tanja Lange, Michiel Kosters, Christiane Peters, Marco Streng

Make sure to see this page for more links.

Hyperbolic geometry

cool hyperbolic picture 1 cool hyperbolic picture 2

Related tickets

Created tickets:


Ticket #4000 on rational polynomials... QQ[x] via FLINT

Sage on GPU's

ZZ[x] factoring in FLINT, plus LLL

Swinnerton-Dyer Polynomials

See trac 9492.

sage: a = sqrt(2)+sqrt(3)+sqrt(5)+sqrt(7)+sqrt(11)
sage: f = algebraic_dependency(a.numerical_approx(10000),32)
sage: b = a.numerical_approx(100000)
sage: time f(b)

# Lift a padic `x` to ZZ, but centered around zero.
def centerlift(x):
    modulus = x.parent().prime_pow(x.precision_absolute())
    z = ZZ(x);
    if (2*z > modulus):
        z -= modulus
    return z

# L = list of numbers you want to take the square root of.
# bound = bound on the absolute value of the coefficients of
# the resulting polynomial.
def swinnerton_dyer(L, bound):
    for p in Primes():
        if all([gcd(p,s) == 1 and is_square(Mod(s,p)) for s in L]):
    prec = ceil(log(bound)/log(p))
    print "Using p =", p
    print "Precision:", p, "^", prec
    K = Qp(p, prec, print_mode="terse", print_pos=False)
    sqrts = [sqrt(K(s),extend=False) for s in L]
    n = len(L)
    padic_roots = []
    for k in range(0, 2^n):
        binary = ZZ(k).digits(base=2,padto=n)
        root = sum([sqrts[i]*(binary[i]*2-1) for i in range(0,n)])
    t = polygen(K)
    pol_padic = prod([t - r for r in padic_roots])
    coeffs_ZZ = [centerlift(c) for c in pol_padic.list()]
    max_coeff = max([abs(c) for c in coeffs_ZZ])
    print "Largest coefficient:", p, "^", ceil(log(max_coeff)/log(p))
    return PolynomialRing(ZZ, names='t')(coeffs_ZZ)

# Example:
time swinnerton_dyer([2,3,5,7,11,13,17,19,23,29], 2^4000)   # bound is heuristic

MPIR projects

Ticket #4260 - Sage + Linbox

Sage Notebook in the classroom

Specific Projects

General framework for the factor(n) command

Solving Conics

Sums of Squares

Porting ECHIDNA code from MAGMA

days23/CodingProjects (last edited 2010-08-31 23:54:42 by was)