Sage 9.6 Release Tour

Sage 9.6 was released on May 15, 2022.

A total of 83 people were involved as authors or reviewers of code contributions to Sage 9.6.

Here is an overview of some of the main changes in this version.

User interface, plotting and graphics

JupyterLab 3.3

JupyterLab, the latest web-based interactive development environment for notebooks, code, and data, is slated to replace the now-classic Jupyter notebook interface. The version of JupyterLab in the Sage distribution has been upgraded to the major new version 3.3. #32069, #33607

After ./sage -i jupyterlab_widgets, you can run it using

./sage -n jupyterlab

Also two new interface variants are provided:

./sage -n nbclassic


./sage -n retrolab

LaTeX displays in JupyterLab

Users of Sage in JupyterLab got used to expressions displayed at center in the LaTeX display mode. For compatibility with displays in classic Jupyter, we decided to change the behavior so that now expressions are displayed aligned left by default.

If you belong to the minority preferring centered displays, you can set your preference by

dm = get_display_manager()   
dm.preferences.align_latex = 'center'  # or 'left'

in the ~/.sage/init.sage script.

Interactive graph editing with phitigra

With the new optional package phitigra (use ./sage -i phitigra to install), graphs can be edited by interactively placing vertices, edges, etc. This works both in the classic Jupyter notebook and in JupyterLab. It can also be used to make animations (see the demo notebook at for examples). Done in #30540 and #33639.

Hyperbolic plots

  • Added the ability to choose the hyperbolic model for hyperbolic plots. #22081

Graphics with TikZ

The TikzPicture module which was developed in the slabbe package for more than 5 years is now in Sage. This was done in ticket #20343. The module is within the new file sage/misc/ and its documentation in the reference manual is available here: Below are some usage examples.

First example shows that it takes any tikz picture string as input:

sage: from sage.misc.latex_standalone import TikzPicture
sage: s = '\\begin{tikzpicture}\n\\draw[->,green,very thick](0,0) -- (1,1);\\end{tikzpicture}'
sage: t = TikzPicture(s)
sage: t        # in Jupyter, rich representation will show the image instead
\draw[->,green,very thick](0,0) -- (1,1);\end{tikzpicture}
sage: path_to_file = t.pdf() # and opens the image in a viewer

Of course, conversion to pdf format necessitates pdflatex or lualatex. If lualatex is available it uses it in preference to pdflatex because it handles better the very big pictures in terms of memory limits.

One can provide a local filename to save to, or convert the image to other formats (using pdftocairo or imagemagick external packages):

sage: path_to_file = t.pdf('file.pdf')  # when providing a filename, it just saves 
                                        # the file locally, does not open in a viewer
sage: path_to_file = t.png() # conversion to png
sage: path_to_file = t.svg() # to svg
sage: path_to_file = t.tex() # print the tex source to a file

Another example with graphs where additional usepackage are necessary to compile the image correctly:

sage: from sage.misc.latex_standalone import TikzPicture
sage: g = graphs.PetersenGraph()
sage: t = TikzPicture(latex(g), standalone_config=["border=4mm"], usepackage=['tkz-graph'])
sage: t        # in Jupyter, rich representation will show the image instead
65 lines not printed (3695 characters in total).
Use print to see the full content.
sage: _ = t.pdf()               # or t.png() or t.svg()

sage: from sage.misc.latex_standalone import TikzPicture
sage: V = [[1,0,1],[1,0,0],[1,1,0],[0,0,-1],[0,1,0],[-1,0,0],[0,1,1],[0,0,1],[0,-1,0]]
sage: P = Polyhedron(vertices=V).polar()
sage: s = P.projection().tikz([674,108,-731],112)
sage: t = TikzPicture(s)
sage: _ = t.pdf()               # or t.png() or t.svg()

The module also contains a class Standalone, from which the class TikzPicture inherits:

sage: from sage.misc.latex_standalone import Standalone
sage: s = Standalone('Hello World', usepackage=['amsmath'], standalone_config=['beamer=true','border=1mm'])
sage: s        # in Jupyter, rich representation will show the image instead
Hello World
sage: _ = s.pdf()               # or s.png() or s.svg()

Another example using Standalone with a tableau:

sage: P = Permutations(10)
sage: p = P.random_element()
sage: p
[3, 10, 1, 9, 5, 6, 7, 2, 8, 4]
sage: t = p.to_tableau_by_shape([3,3,3,1])
sage: t
[[2, 8, 4], [5, 6, 7], [10, 1, 9], [3]]
sage: s = Standalone(latex(t), standalone_config=["border=1mm"])
sage: s
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}} }
sage: _ = s.pdf()               # or s.png() or s.svg()

In a next step, a method tikz() will be added to graphs, polytopes, posets, etc. to return an object of type TikzPicture see #33002.

Complex plots

The complex plotting package phase_mag_plot has been incorporated into Sage. Now complex_plot allows contouring, tiling, and matplotlib-compatible colormaps. This was added in ticket #33416.

To use a colormap, one can pass in a string as in

sage: complex_plot((x - 5)*sqrt(x), (-10, 10), (-10, 10), cmap='twilight')

Contouring or tiling are enabled through keyword options. To look smooth, it's typically necessary to plot the function on additional points through the use of plot_points. This looks like

sage: complex_plot((x - 5)*sqrt(x), (-10, 10), (-10, 10), cmap='twilight', plot_points=500, contoured=True)

sage: complex_plot((x - 5)*sqrt(x), (-10, 10), (-10, 10), cmap='twilight', plot_points=500, tiled=True)

Linear algebra

NumPy integration

The new classes Matrix_numpy_integer_dense and Vector_numpy_integer_dense implement matrices and vectors with 64-bit integer entries backed by numpy arrays. #32465.

As a first application, several methods of GenericGraph that return matrices, such as adjacency_matrix, now accept keyword arguments that can select the matrix implementation. #33377, #33387, #33388, #33389

sage: graphs.PathGraph(5).adjacency_matrix(sparse=False, implementation='numpy')
[0 1 0 0 0]
[1 0 1 0 0]
[0 1 0 1 0]
[0 0 1 0 1]
[0 0 0 1 0]
sage: type(_)
<class 'sage.matrix.matrix_numpy_integer_dense.Matrix_numpy_integer_dense'>

CombinatorialFreeModule improvements

Performing sums and similar constructions for CombinatorialFreeModule have been made faster. #33267


SymPy 1.10.1

SymPy has been upgraded to version 1.10.1 (release notes). #33398, #33547, #33584


Sage 9.6 defines a new class ImageSet. #32121

sage: ImageSet(sin,, pi/4))
Image of (0, 1/4*pi) by The map sin from (0, 1/4*pi)
sage: _.an_element()
1/2*sqrt(-sqrt(2) + 2)

sage: sos(x,y) = x^2 + y^2; sos
(x, y) |--> x^2 + y^2
sage: ImageSet(sos, ZZ^2)
Image of
 Ambient free module of rank 2 over the principal ideal domain Integer Ring by
 The map (x, y) |--> x^2 + y^2 from Vector space of dimension 2 over Symbolic Ring
sage: _.an_element()
sage: ImageSet(sos, Set([(3, 4), (3, -4)]))
Image of {...(3, -4)...} by
 The map (x, y) |--> x^2 + y^2 from Vector space of dimension 2 over Symbolic Ring
sage: _.an_element()

The new class mirrors and translates to SymPy's ImageSet:

sage: from sage.sets.image_set import ImageSet
sage: S = ImageSet(sin,, pi/4)); S
Image of (0, 1/4*pi) by The map sin from (0, 1/4*pi)
sage: S._sympy_()
ImageSet(Lambda(x, sin(x)),, pi/4))

Most methods of ImageSet are actually provided by its base class, the new class ImageSubobject. For all morphisms in the Sets category, there is now a default method image, which constructs an instance of either ImageSubobject or ImageSet.

Orthogonal polynomials

Three classes of classical (discrete) orthogonal polynomials in the Askey scheme have been added: #33393

  • Krawtchouk polynomials
  • Hahn polynomials
  • Meixner polynomials

Spherical harmonics

Various issues regarding spherical harmonics have been fixed (#33117, #33501). In particular, the Condon-Shortley phase has been added, so that Sage's spherical harmonics now agree with those of Wikipedia, SymPy, SciPy and Mathematica. For instance

sage: theta, phi = var('theta phi')
sage: spherical_harmonic(1, 1, theta, phi)

This clearly agrees with SymPy's spherical harmonics:

sage: from sympy import Ynm
sage: Ynm(1, 1, theta, phi).expand(func=True) 

Polyhedral geometry and linear programming

polymake 4.6

polymake, a comprehensive system for computations in polyhedral geometry, tropical geometry, etc., has been upgraded to version 4.6 (release notes). #33251


The new optional package CyLP (./sage -i cylp) provides a detailed interface to Clp, the COIN-OR linear programming solver, and Cbc, the COIN-OR branch-and-cut solver for mixed-integer linear programs. #33847

In a future version of Sage, CyLP is intended to provide a replacement for the Sage-specific backend interface to Clp and Cbc, sage-numerical-backends-coin; see Meta-ticket #26511.

Equivariant Ehrhart theory



Improved Manifold constructor

The structure parameter of the Manifold constructor has new, more convenient defaulting behavior. #33001

When parameters such as diff_degree or metric_name are given, the implied structure is selected:

sage: M = Manifold(3, 'M', diff_degree=0); M
3-dimensional topological manifold M
sage: M = Manifold(3, 'M', diff_degree=2); M
3-dimensional differentiable manifold M
sage: M = Manifold(3, 'M', metric_name='g'); M
3-dimensional Riemannian manifold M

Symplectic manifolds

Symplectic structures have been added to Sage (#30362).

The standard symplectic vector space can be obtained as follows:

sage: M.<q, p> = manifolds.StandardSymplecticSpace(2, symplectic_name='omega')
sage: omega = M.symplectic_form()
saga: omega.display()
omega = -dq∧dp

To use an existing 2-form as a symplectic form use the new wrap method.

sage: from sage.manifolds.differentiable.symplectic_form import SymplecticForm
sage: M = manifolds.Sphere(2, coordinates='stereographic')
sage: vol_form = M.induced_metric().volume_form()
sage: omega = SymplecticForm.wrap(vol_form, 'omega', r'\omega')
sage: omega.display()
omega = -4/(y1^4 + y2^4 + 2*(y1^2 + 1)*y2^2 + 2*y1^2 + 1) dy1∧dy2

Currently, the following operations from symplectic geometry are supported:

  • Musical isomorphism (flat/sharp) between vector fields and 1-forms.
  • Poisson tensor
  • Liouville volume form
  • Poisson bracket of functions
  • Hamiltonion vector fields
  • Symplectic Hodge dual of a differential form

Projective spaces

Real projective spaces have been added to the manifold catalog (#33221). For example, one can construct the real projective plane.

sage: RP2 = manifolds.RealProjectiveSpace(); RP2
2-dimensional topological manifold RP2
sage: latex(RP2)

There are three charts. Considering an immersion in three-dimensional Euclidean space, the chart corresponds to a choice of one (out of three possible) coordinates to be always nonzero. The coordinates listed are the two other coordinates assuming that the nonzero coordinate is always one.

sage: C0, C1, C2 = RP2.top_charts()
sage: p = RP2.point((2,0), chart = C0)
sage: q = RP2.point((0,3), chart = C0)
sage: p in C0.domain()
sage: p in C1.domain()
sage: C1(p)
(1/2, 0)
sage: p in C2.domain()
sage: q in C0.domain()

The point q looks like (1,0,3) in ambient Euclidean space, so it is not in the domain of the chart C1. It also has the form (1/3,0,1) when considered in the chart C2.

sage: q in C1.domain()
sage: q in C2.domain()
sage: C2(q)
(1/3, 0)

If both coordinates in a local chart are nonzero then that point is in the domain of all charts. The change of coordinates is found by normalizing the appropriate chart. So for example, the point (1, 2, 3) is the same as the point (1/2, 1, 3/2) is the same as the point (1/3, 2/3, 1), which is reflected by defining r to be a point in the default chart C0.

sage: r = RP2.point((2,3))
sage: r in C0.domain() and r in C1.domain() and r in C2.domain()
sage: C0(r) # corresponding to (1, 2, 3)
(2, 3)
sage: C1(r) # corresponding to (1/2, 1, 3/2)
(1/2, 3/2)
sage: C2(r) # corresponding to (1/3, 2/3, 1)
(1/3, 2/3)

Internal code improvements and bug fixes

Some performance improvements have been implemented (#33110):

  • use the first Bianchi identity in the computation of the Riemann tensor
  • compute volume forms with contravariant indices only as needed
  • no try to simplify trivial expressions consisting only of a single number or symbolic variable

Some bugs have been corrected: #32953, #33399, #33780.


lrcalc 2.1

lrcalc, Anders Buch's Littlewood-Richardson Calculator, has been upgraded to the major new version 2.1 changelog. #31355

Finitely presented modules over graded algebras

Sage 9.6 allows the construction of finitely presented modules over graded algebras, even algebras which are infinite and/or noncommutative like the mod p Steenrod algebra. Some homological algebra is implemented in general, and more tools are implemented specifically for modules over the Steenrod algebra (#32505, #30680).

sage: from sage.modules.fp_graded.module import FPModule
sage: E.<x,y> = ExteriorAlgebra(QQ)
# M has one generator g in degree 0 and two relations, x*g and y*g.
# That is, M is QQ as a trivial E-module.
sage: M = FPModule(E, [0], [[x], [y]])
# Free resolution:
sage: M.resolution(3)
[Module morphism:
   From: Free graded left module on 1 generator over 
          The exterior algebra of rank 2 over Rational Field
   To:   Finitely presented left module on 1 generator and 2 relations over 
          The exterior algebra of rank 2 over Rational Field
   Defn: g[0] |--> g[0],
 Module morphism:
   From: Free graded left module on 2 generators over 
          The exterior algebra of rank 2 over Rational Field
   To:   Free graded left module on 1 generator over 
          The exterior algebra of rank 2 over Rational Field
   Defn: g[1, 0] |--> x*g[0]
         g[1, 1] |--> y*g[0],
 Module morphism:
   From: Free graded left module on 3 generators over 
          The exterior algebra of rank 2 over Rational Field
   To:   Free graded left module on 2 generators over 
          The exterior algebra of rank 2 over Rational Field
   Defn: g[2, 0] |--> x*g[1, 0]
         g[2, 1] |--> y*g[1, 0] + x*g[1, 1]
         g[2, 2] |--> y*g[1, 1],
 Module morphism:
   From: Free graded left module on 4 generators over 
          The exterior algebra of rank 2 over Rational Field
   To:   Free graded left module on 3 generators over 
          The exterior algebra of rank 2 over Rational Field
   Defn: g[3, 0] |--> x*g[2, 0]
         g[3, 1] |--> y*g[2, 0] + x*g[2, 1]
         g[3, 2] |--> y*g[2, 1] + x*g[2, 2]
         g[3, 3] |--> y*g[2, 2]]

There is a new thematic tutorial providing many details and examples.

Miscellaneous improvements

  • Ideal membership over quotient rings can now be decided (by reducing to ideal membership in the parent ring). #33237
  • Iterating over (some) infinite modules (including ℤn) now enumerates the entire module, in a "natural" order. #33287
  • BinaryQF.solve_integer() now also works for quadratic forms of square discriminant. #33026
  • Quaternion fractional ideals (including orders) now support the usual operations (e.g., a*I, I*a, I+J). #32264
  • AdditiveAbelianGroupWrapper now exposes .discrete_log() for (multi-dimensional) logarithms in finite abelian groups. #32384
  • Graded submodules of graded modules now know they are graded (with respect to the ambient grading); similarly for filtered submodules. #33321
  • Polynomials now evaluate faster on monomial inputs. #33165
  • Implement specialized code for summing terms and monomials in CombinatorialFreeModule. #33267
  • Improvements and fixes to skew_by() in symmetric functions. #33313
  • Attempt to invert elements generically in a finite dimensional algebra. #33250
  • Tensor products of finite dimensional modules know they are finite dimensional (Sage does not currently have any structure for infinite tensor products, which can have some subtleties). #30252
  • Improved coercions and conversions with Laurent polynomials and their fraction field. #31320 #33477
  • Faster evaluation of univariate polynomials with monomials. #33165

Number theory

Elliptic curves

  • Elliptic-curve DLP and Weil pairings over finite fields are now much faster (thanks to PARI). #33121
  • Scalar multiplication on elliptic curves over finite fields is now significantly faster (thanks to PARI). #33147
  • Computing the Weierstraß ℘ function of an elliptic curve is now significantly faster (thanks to PARI). #33223
  • Classes used by the Monsky-Washnitzer curves now use the new coercion system. #33525 #33576


Optimizations to SBox. #25633

Package upgrades

For a list of all packages and their versions, see

Python 3.10

Sage 9.6 continues to support system Python 3.7.x to 3.10.x. If no suitable version of Python is installed in the system, Sage will install its own copy of Python. Sage now ships Python 3.10.3 for this purpose. #30767, #33512

FLINT 2.8.x and arb 2.22.x

FLINT has been upgraded from 2.7.1 to 2.8.4.#32211

The FLINT 2.8 series brings major new algorithms and general speedups (release notes).

Note that Sage accepts system installations of FLINT >= 2.6.x. Users on older distributions who want to benefit from the speed ups in FLINT 2.8.x may want to use ./configure --without-system-flint.

Meta-ticket #31408 tracks the effort to make use of new features added in recent FLINT releases in the Sage library.

arb has been upgraded from 2.19.0 to 2.22.1. #32211, #33189

The 2.20, 2.21, and 2.22 series have brought major new algorithms and other improvements (changelog).

igraph 0.9.x

The igraph library and its Python interface (now also just called igraph) have been upgraded to versions 0.9.7/0.9.9. #32510, #33526

For developers: Refactoring and modularization

See also Meta-ticket #29705


The Sage library interfaces to some external non-Python packages by running an executable program in a separate process. The package may either be available from a system installation, or the Sage distribution may have installed the package in the SAGE_LOCAL prefix hierarchy. The main sage script sets up various environment variables before starting the Python interpreter; in particular, it sets PATH to include SAGE_LOCAL/bin, which ensures that the installed executables are found.

In Sage 9.6, we have changed most calls to executables so that they no longer depend on the environment variable PATH being set in this way. Every executable program is represented by an instance of sage.features.Executable. Its method absolute_filename explicitly searches inside SAGE_LOCAL/bin (in installations with SAGE_LOCAL) before it depends on PATH. #31292, #31296, #32645, #32893, #33440, #33465, #33467

Preparations for PEP 420 implicit namespace packages

The Sage doctester is now prepared for namespace packages. #33033

To reduce runtime dependencies, many imports from sage.graphs.all, sage.interfaces.all, sage.misc.all, sage.libs.all, sage.rings.all have been replaced by more specific imports; and module-level imports from sage.plot have been changed to either use lazy_import or have been moved inside methods. #30582, #32847, #32989, #32999, #33000, #33007, #33146, #33199, #33466, #33468

Lazy imports of classes now support "isinstance"

No Python object is an instance of a class that cannot be imported from the module that defines it. The new special method LazyImport.__instancecheck__ now just returns False in this case. #33017

This provides a convenient pattern for writing modularized code when creating an abstract base class for "isinstance" testing is not justified.

sage: from sage.schemes.generic.scheme import Scheme
sage: sZZ = Scheme(ZZ)
sage: lazy_import('xxxx_does_not_exist', 'Pluffe')
sage: isinstance(sZZ, (Scheme, Pluffe))
sage: isinstance(ZZ, (Scheme, Pluffe))

Likewise, no class is a subclass of a class that cannot be imported from the module that defines it; so the new special method LazyImport.__subclasscheck__ implements the same logic.

sage.geometry.polyhedron.base reorganized

The module sage.geometry.polyhedron.base has been split into several modules, grouping the methods of class Polyhedron_base according to their topic and runtime dependencies on other parts of Sage. #32651

New developer tools

Pre-built Docker containers on

Our portability CI on GitHub Actions builds Docker images for all tested Linux platforms (and system package configurations) and makes them available on GitHub Packages ( #30933

Since 9.6.beta1, the image version corresponding to the latest development release receives the additional Docker tag dev, see for example the Docker image for ubuntu-impish-standard. Thus, for example, the following command will work:

$ docker run -it bash

Images whose names end with the suffix -with-targets-optional are the results of full builds and a run of make ptest. They also contain a copy of the source tree and the full logs of the build and test.

Smaller images corresponding to earlier build stages are also available:

  • -with-system-packages provides a system installation with system packages installed, no source tree,
  • -configured contains a partial source tree and has completed bootstrap and the configure,
  • -with-targets-pre contains the full source tree and a full installation of all non-Python packages,
  • -with-targets contains the full source tree and a full installation of Sage, including the HTML documentation, but make ptest has not been run yet.

Sage development in the cloud with Gitpod

Gitpod is a service that provides a development environment in the cloud based on VS Code. It is free to use for up to 50 hours per month. Sage now includes a configuration for Gitpod; see the new section Setting up your workspace in the Sage Developer's Guide. #33103, #33589

To launch Gitpod on a branch of a Trac ticket, you can use the new badge in the ticket box.

Alternatively, prepend any repository URL with; for example, opens a development environment containing a prebuilt copy of Sage corresponding to the develop branch.

Builds and checks of ticket branches on GitHub Actions

Next to the familiar patchbot badges, each ticket now has badges linking to tests that run on GitHub Actions.

Linting workflow (pycodestyle, relint)

A linting workflow on GitHub Actions runs on all pushes to a branch on Trac. It checks that the code of the current branch adheres to the style guidelines using pycodestyle (in the pycodestyle-minimal configuration) and relint.

In order to see details when it fails, you can click on the badge and then select the most recent workflow run.

Sage 9.6 includes again many improvements to the coding style checked by pycodestyle. #32978, #32979, #33095, #33102, ...

Consequently, the pycodestyle-minimal configuration has been extended to enforce E111, E401, E701, E702, E703, W605, E711, E712, E713, E721, E722.

Incremental build and test

The build & test workflow on GitHub Actions builds Sage for the current branch (incrementally on top of the system packages of the develop branch) and runs the test.

Details are again available by clicking on the badge.

Note that in contrast to the patchbot, the ticket branch is not merged into the current beta version.

Documentation build

The build documentation workflow on GitHub Actions builds the HTML documentation for the current branch.

If you click on the badge, you get the HTML output of the successful run. The idea is to use this to easily inspect changes to the documentation without the need to locally rebuild the docs yourself. If the doc build fails, you can go to the Actions tab of sagemath/sagetrac-mirror repo and choose the particular branch to see what went wrong.

The idea is that these three status badges complement the existing patchbots (and maybe even replace them in the future). In particular, they are supposed to always be green.

Test coverage analysis with codecov

A detailed coverage analysis of the Sage library is now available at #33355

sage --pytest

After the optional package pytest is installed (./sage -i pytest), a new command ./sage --pytest is available, which runs pytest on the Sage library sources. #33572

Also the Sage doctester (./sage -t or ./sage -tox -e doctest) invokes pytest. This functionality has been improved in Sage 9.6. #31924, #32975, #33560

sage -t --baseline-stats-path


Sage patchbot on GitHub Actions

The Sage patchbot can now be run on GitHub Actions, on top of any of the Linux platforms for which we have prebuilt Docker images.#33253

New front page

The front page of has been reorganized. #33725

Availability of Sage 9.6 and installation help

Sage 9.6 was released on 2022-05-15.

Please read the updated SageMath installation guide

  • It now provides a decision tree that guides users and developers to a type of installation suitable for their system and their needs.
  • The section on conda-forge has been updated and now includes (still experimental) instructions for Sage development on top of conda-forge.


The Sage source code is available in the sage git repository.

SageMath 9.6 supports all platforms that were supported by Sage 9.5 and adds support for building on distributions that use the GCC 12 series (fedora-36). #33187

(On platforms marked with the superscript ⁺, installing optional packages is not supported in Sage 9.6; and support for these platforms will be removed in Sage 9.7; see #32074. Upgrade to a newer version of the distribution or at least upgrade the toolchain (gcc, binutils).)

Availability as binaries

  • Binary package for 9.6 on macOS from the 3-manifolds project
    • separate disk images for Intel (x86_64) and Apple Silicon (M1, arm64)
    • also includes many optional Sage packages
  • CoCalc-Docker from the CoCalc project
    • A large image that has Sage, Jupyter, Pluto.jl, VS Code, a web-based X11 server, LaTeX, SageTex?, a new collaborative Whiteboard with Sage code cells
    • But it's big -- 6GB compressed (and 25GB uncompressed) -- so may or may not be of use depending on what you're doing...


See in the source distribution for installation instructions.

More details

Last modified 3 days ago Last modified on 05/20/22 08:08:55

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