Sage 4.1.1 Release Tour

Algebra

• FIXME: summarize #6510

Basic Arithmetic

• FIXME: summarize #6159

• FIXME: summarize #2737

Calculus

• FIXME: summarize #5996

Combinatorics

• FIXME: summarize #6519

• FIXME: summarize #6621

• FIXME: summarize #5790

Cryptography

• FIXME: summarize #6454

Documentation

• FIXME: summarize #4460

Elliptic Curves

• #6381 (bug in integral_points when rank is large):

The function integral_x_coords_in_interval() for finding all integral points on an elliptic curve defined over the rationals whose x-coordinate lies in an interval is now more efficient when the interval is large.

• FIXME: summarize #6407

Graphics

• FIXME: summarize #6098

• FIXME: summarize #5649

• FIXME: summarize #5651

Graph Theory

• FIXME: summarize #6355

• FIXME: summarize #6540

• FIXME: summarize #6552

• FIXME: summarize #6578

• FIXME: summarize #5793

Interfaces

• FIXME: summarize #6395

• FIXME: summarize #6591

Linear Algebra

• FIXME: summarize #5081

• FIXME: summarize #6553

• FIXME: summarize #6554

• FIXME: summarize #6301

Modular Forms

• FIXME: summarize #6606

• FIXME: summarize #6071

• FIXME: summarize #6534

Notebook

• FIXME: summarize #5653

Number Theory

• #6457 (Intersection of ideals in a number field)

Intersection of ideals in number fields is now implemented.

• FIXME: summarize #6045

• FIXME: summarize #6396

• FIXME: summarize #6458

Numerical

• FIXME: summarize #6506

• FIXME: summarize #4571

Packages

• FIXME: summarize #6558

• FIXME: summarize #6380

• FIXME: summarize #6443

• FIXME: summarize #6445

• FIXME: summarize #6451

• FIXME: summarize #6453

• FIXME: summarize #6528

• FIXME: summarize #6143

• FIXME: summarize #6438

• FIXME: summarize #6493

• FIXME: summarize #6563

• FIXME: summarize #6602

• FIXME: summarize #6302

• new optional package p_group_cohomology (Simon A. King, David J. Green)

  • Compute the cohomology ring with coefficients in GF(p) for any finite p-group, in terms of a minimal generating set a minimal set of algebraic relations. We use Benson's criterion to prove the completeness of the ring structure.
  • Compute depth, dimension, Poincare series and a-invariants of the cohomology rings.
  • Construct induced homomorphisms.
  • The package includes a list of cohomology rings for all groups of order 64.
  • With the package, the cohomology for all groups of order 128 and for the Sylow 2-subgroup of the third Conway group (order 1024) was computed for the first time. The result of these and many other computations (e.g., all but 6 groups of order 243) is accessible in a repository on sage.math.

  Examples:

  • Data that are included with the package:

    sage: from pGroupCohomology import CohomologyRing
    sage: H = CohomologyRing(64,132) # this is included in the package, hence, the ring structure is already there
    sage: print H
    
    Cohomology ring of Small Group number 132 of order 64 with coefficients in GF(2)
    
    Computation complete
    Minimal list of generators:
    [a_2_4, a 2-Cochain in H^*(SmallGroup(64,132); GF(2)),
     c_2_5, a 2-Cochain in H^*(SmallGroup(64,132); GF(2)),
     c_4_12, a 4-Cochain in H^*(SmallGroup(64,132); GF(2)),
     a_1_0, a 1-Cochain in H^*(SmallGroup(64,132); GF(2)),
     a_1_1, a 1-Cochain in H^*(SmallGroup(64,132); GF(2)),
     b_1_2, a 1-Cochain in H^*(SmallGroup(64,132); GF(2))]
    Minimal list of algebraic relations:
    [a_1_0*a_1_1,
     a_1_0*b_1_2,
     a_1_1^3+a_1_0^3,
     a_2_4*a_1_0,
     a_1_1^2*b_1_2^2+a_2_4*a_1_1*b_1_2+a_2_4^2+c_2_5*a_1_1^2]
    sage: H.depth()
    2
    sage: H.a_invariants()
    [-Infinity, -Infinity, -3, -3]
    sage: H.poincare_series()
    (-t^2 - t - 1)/(t^6 - 2*t^5 + t^4 - t^2 + 2*t - 1)

  • Data from the repository on sage.math:

    sage: H = CohomologyRing(128,562)
    sage: len(H.gens())
    35
    sage: len(H.rels())
    486
    sage: H.depth()
    1
    sage: H.a_invariants()
    [-Infinity, -4, -3, -3]
    sage: H.poincare_series()
    (t^14 - 2*t^13 + 2*t^12 - t^11 - t^10 + t^9 - 2*t^8 + 2*t^7 - 2*t^6 + 2*t^5 - 2*t^4 + t^3 - t^2 - 1)/(t^17 - 3*t^16 + 4*t^15 - 4*t^14 + 4*t^13 - 4*t^12 + 4*t^11 - 4*t^10 + 4*t^9 - 4*t^8 + 4*t^7 - 4*t^6 + 4*t^5 - 4*t^4 + 4*t^3 - 4*t^2 + 3*t - 1)

  • Some computation from scratch, involving different ring presentations and induced maps:

    sage: tmp_root = tmp_filename()
    sage: CohomologyRing.set_user_db(tmp_root)
    sage: H0 = CohomologyRing.user_db(8,3,websource=False)
    sage: print H0
    
    Cohomology ring of Dihedral group of order 8 with coefficients in GF(2)
    
    Computed up to degree 0
    Minimal list of generators:
    []
    Minimal list of algebraic relations:
    []
    
    sage: H0.make()
    sage: print H0
    
    Cohomology ring of Dihedral group of order 8 with coefficients in GF(2)
    
    Computation complete
    Minimal list of generators:
    [c_2_2, a 2-Cochain in H^*(D8; GF(2)),
     b_1_0, a 1-Cochain in H^*(D8; GF(2)),
     b_1_1, a 1-Cochain in H^*(D8; GF(2))]
    Minimal list of algebraic relations:
    [b_1_0*b_1_1]
    
    sage: G = gap('DihedralGroup(8)')
    sage: H1 = CohomologyRing.user_db(G,GroupName='GapD8',websource=False)
    sage: H1.make()
    sage: print H1  # the ring presentation is different ...
    
    Cohomology ring of GapD8 with coefficients in GF(2)
    
    Computation complete
    Minimal list of generators:
    [c_2_2, a 2-Cochain in H^*(GapD8; GF(2)),
     b_1_0, a 1-Cochain in H^*(GapD8; GF(2)),
     b_1_1, a 1-Cochain in H^*(GapD8; GF(2))]
    Minimal list of algebraic relations:
    [b_1_1^2+b_1_0*b_1_1]
    
    sage: phi = G.IsomorphismGroups(H0.group())
    sage: phi_star = H0.hom(phi,H1)
    sage: phi_star_inv = phi_star^(-1) # ... but the rings are isomorphic
    sage: [X==phi_star_inv(phi_star(X)) for X in H0.gens()]
    [True, True, True, True]
    sage: [X==phi_star(phi_star_inv(X)) for X in H1.gens()]
    [True, True, True, True]

  • An example with an odd prime:

    sage: H = CohomologyRing(81,8) # this needs to be computed from scratch
    sage: H.make()
    sage: H.gens()
    
    [1,
     a_2_1, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_2_2, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)),
     b_2_0, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_4_1, a 4-Cochain in H^*(SmallGroup(81,8); GF(3)),
     b_4_2, a 4-Cochain in H^*(SmallGroup(81,8); GF(3)),
     b_6_3, a 6-Cochain in H^*(SmallGroup(81,8); GF(3)),
     c_6_4, a 6-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_1_0, a 1-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_1_1, a 1-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_3_2, a 3-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_5_2, a 5-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_5_3, a 5-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_7_5, a 7-Cochain in H^*(SmallGroup(81,8); GF(3))]
    sage: len(H.rels())
    59
    sage: H.depth()
    1
    sage: H.a_invariants()
    [-Infinity, -3, -2]
    sage: H.poincare_series()
    (t^4 - t^3 + t^2 + 1)/(t^6 - 2*t^5 + 2*t^4 - 2*t^3 + 2*t^2 - 2*t + 1)

Symbolics

• FIXME: summarize #6195

• FIXME: summarize #6243