Sage 4.3.1 Release Tour

Major features

• Substantial work towards a complete SPARC Solaris 10 port. This is due to the hard work of David Kirkby. The relevant tickets include #6595, #7138, #7162, #7505, #7817.

• We're moving closer towards a FreeBSD port, thanks to the work of Peter Jeremy at ticket #7825.

Basic arithmetics

• Implement conjugate() for RealDoubleElement #7834 (Dag Sverre Seljebotn) --- New method conjugate() in the class RealDoubleElement of the module sage/rings/real_double.pyx for returning the complex conjugate of a real number. This is consistent with conjugate() methods in ZZ and RR. For example,

  sage: ZZ(5).conjugate()
  5
  sage: RR(5).conjugate()
  5.00000000000000
  sage: RDF(5).conjugate()
  5.0

Combinatorics

• #7754 (Nicolas M. Thiéry)

Elliptic curves

• Two-isogeny descent over QQ natively using ratpoints #6583 (Robert Miller) --- New module sage/schemes/elliptic_curves/descent_two_isogeny.pyx for descent on elliptic curves over QQ with a 2-isogeny. The relevant user interface function is two_descent_by_two_isogeny() that takes an elliptic curve E with a two-isogeny phi : E --> E' and dual isogeny phi', runs a two-isogeny descent on E, and returns n1, n2, n1' and n2'. Here, n1 is the number of quartic covers found with a rational point and n2 is the number which are ELS. Here are some examples illustrating the use of two_descent_by_two_isogeny():

  sage: from sage.schemes.elliptic_curves.descent_two_isogeny import two_descent_by_two_isogeny
  sage: E = EllipticCurve("14a")
  sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E)
  sage: log(n1, 2) + log(n1_prime, 2) - 2  # the rank
  0
  sage: E = EllipticCurve("65a")
  sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E)
  sage: log(n1, 2) + log(n1_prime, 2) - 2  # the rank
  1
  sage: E = EllipticCurve("1088j1")
  sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E)
  sage: log(n1, 2) + log(n1_prime, 2) - 2  # the rank
  2

  You could also ask two_descent_by_two_isogeny() to be verbose in its computation:

  sage: E = EllipticCurve("14a")
  sage: two_descent_by_two_isogeny(E, verbosity=1)
  2-isogeny
  Results:
  2 <= #E(Q)/phi'(E'(Q)) <= 2
  2 <= #E'(Q)/phi(E(Q)) <= 2
  #Sel^(phi')(E'/Q) = 2
  #Sel^(phi)(E/Q) = 2
  1 <= #Sha(E'/Q)[phi'] <= 1
  1 <= #Sha(E/Q)[phi] <= 1
  1 <= #Sha(E/Q)[2], #Sha(E'/Q)[2] <= 1
  0 <= rank of E(Q) = rank of E'(Q) <= 0
  (2, 2, 2, 2)

• More functions for elliptic curve isogenies #6887 (John Cremona, Jenny Cooley) --- Code for constructing elliptic curve isogenies already existed in Sage 4.1.1. The enhancements here include:

  • For l=2,3,5,7,13 over any field, find all l-isogenies of a given elliptic curve. (These are the l for which X_0(l) has genus 0).

  • Similarly for the remaining l for which l-isogenies exist over QQ.

  • Given an elliptic curve over QQ, find the whole isogeny class in a robust manner.

  • Testing if two curves are isogenous at least over QQ.

  The relevant use interface method is isogenies_prime_degree() in the class EllipticCurve_field of the module sage/schemes/elliptic_curves/ell_field.py. Here are some examples showing isogenies_prime_degree() in action. Examples over finite fields:

  sage: E = EllipticCurve(GF(next_prime(1000000)), [7,8])
  sage: E.isogenies_prime_degree()
  [Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
  sage: E.isogenies_prime_degree(13)
  [Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]

  Examples over number fields (other than QQ):

  sage: QQroot2.<e> = NumberField(x^2 - 2)
  sage: E = EllipticCurve(QQroot2, [1,0,1,4,-6])
  sage: E.isogenies_prime_degree(2)
  [Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-36)*x + (-70) over Number Field in e with defining polynomial x^2 - 2]
  sage: E.isogenies_prime_degree(3)
  [Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-171)*x + (-874) over Number Field in e with defining polynomial x^2 - 2, Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-128/3)*x + 5662/27 over Number Field in e with defining polynomial x^2 - 2]

Graph theory

• An interactive graph editor #1321 (Radoslav Kirov, Mitesh Patel) --- Embed an interactive graph editor into the notebook. The following screenshot shows a graph editor for playing around with the complete graph on 5 vertices:

  

• Breadth/depth first searches and basic connectivity for c_graphs #7724 (Nathann Cohen, Yann Laigle-Chapuy) --- Implementation of the following methods for the class CGraphBackend in the module sage/graphs/base/c_graph.pyx:

  • depth_first_search()

  • breadth_first_search()

  • is_connected()

  • is_strongly_connected()

  In some cases, the c_graphs implementation of these methods provides a 2x speed improvement:

  sage: g = graphs.RandomGNP(1000, 0.01)
  sage: h = g.copy(implementation="c_graph")
  sage: %timeit list(g.depth_first_search(0));
  100 loops, best of 3: 8.17 ms per loop
  sage: %timeit list(h.depth_first_search(0));
  100 loops, best of 3: 3.29 ms per loop
  sage: 
  sage: %timeit list(g.breadth_first_search(0));
  100 loops, best of 3: 6.48 ms per loop
  sage: %timeit list(h.breadth_first_search(0));
  10 loops, best of 3: 34 ms per loop
  sage: 
  sage: %timeit g.is_connected();
  100 loops, best of 3: 8.47 ms per loop
  sage: %timeit h.is_connected();
  100 loops, best of 3: 3.41 ms per loop
  sage:
  sage: g = g.to_directed()
  sage: h = g.copy(implementation="c_graph")
  sage: %timeit g.is_strongly_connected();
  10 loops, best of 3: 23.5 ms per loop
  sage: %timeit h.is_strongly_connected();
  10 loops, best of 3: 25 ms per loop

• Tower of Hanoi graph #7770 (Rob Beezer) --- The Tower of Hanoi puzzle can be described by a graph whose vertices are possible states of the disks on the pegs, with edges representing legitimate moves of a single disk. The new method HanoiTowerGraph() of the class GraphGenerators in the module sage/graphs/graph_generators.py returns the graph whose vertices are the states of the Tower of Hanoi puzzle, with edges representing legal moves between states. See the documentation of this method for details on interpreting the the possible states of this puzzle. The following screenshot shows all the possible states of an instance of the puzzle with 3 pegs and 3 disks, produced using the following code:

  H = graphs.HanoiTowerGraph(3, 3, positions=False)
  show(H, figsize=[8,8])

  

Linear algebra

• #5174 (John Palmieri)

• #7728 (Dag Sverre Seljebotn)

Miscellaneous

• #6820 (John Palmieri, Mitesh Patel)

• #7482 (William Stein)

• #7514 (William Stein)

Packages

• #7271 (Martin Albrecht)

• #7388 (Robert Miller)

• #7483 (William Stein)

• #7692, #7749 (Steven Sivek)

• #7745 (Karl-Dieter Crisman)

• #7825 (Peter Jeremy)

• #7840 (William Stein)