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

• #1321 (Radoslav Kirov, Mitesh Patel)

• #7724 (Nathann Cohen, Yann Laigle-Chapuy)

• #7770 (Rob Beezer)

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)