AlgebraicField = QQbar is a separate implementation from NumberField. The number field embeddings refer to the latter only.
On Sunday, January 17, 2016 at 6:37:14 PM UTC, William wrote: > > Hi, > > Does anybody understand number field embeddings > > http://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/number_field_morphisms.html > > > and whether the default I in SR (i.e., NumberField(x^2+1)) is equipped > with any embeddings. The above reference manual shows how to make > number fields with embeddings, but I couldn't see how to query a > number field for its embeddings. Obviously, I could figure this out > by reading the source, but I'm don't have time right now. I'm curious > though, since I want to talk about this in my grad course on > Wednesday, and it's relevant to the bug. > > I'm posting worksheets like this for my class: > > > https://cloud.sagemath.com/projects/b85dd1b3-3f5f-4c2d-b503-242612561b9e/files/lectures/2016-01-15/2016-01-15-part0-QQbar.sagews > > > On Sun, Jan 17, 2016 at 10:27 AM, Volker Braun <vbrau...@gmail.com > <javascript:>> wrote: > > SR is special in that it wraps various other rings: > > > > sage: wrapped_qqbar = SR(QQbar(I)) > > sage: wrapped_qqbar.parent() > > Symbolic Ring > > sage: wrapped_qqbar.pyobject() > > 1*I > > sage: wrapped_qqbar.pyobject().parent() > > Algebraic Field > > > > You don't gain anything from wrapping stuff in SR; ideally you can avoid > it > > in your code. > > > > I opened http://trac.sagemath.org/ticket/19904 with a better testcase > for > > the segfault > > > > sage: bool(SR(QQbar(I)) == I) > > terminate called after throwing an instance of 'std::runtime_error' > > what(): > > ------------------------------------------------------------------------ > > > /home/vbraun/Code/sage/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x3da5)[0x7f06b9187da5] > > > > > /home/vbraun/Code/sage/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x3df7)[0x7f06b9187df7] > > > > > /home/vbraun/Code/sage/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x644c)[0x7f06b918a44c] > > > > /lib64/libpthread.so.0(+0x109f0)[0x7f06c8b0e9f0] > > /lib64/libc.so.6(gsignal+0x38)[0x7f06c8068a98] > > /lib64/libc.so.6(abort+0x16a)[0x7f06c806a69a] > > > /home/vbraun/Code/sage/local/lib/libstdc++.so.6(__gnu_cxx::__verbose_terminate_handler()+0x15d)[0x7f06acab2ccd] > > > > > /home/vbraun/Code/sage/local/lib/libstdc++.so.6(+0x5dd46)[0x7f06acab0d46] > > > /home/vbraun/Code/sage/local/lib/libstdc++.so.6(+0x5dd91)[0x7f06acab0d91] > > > /home/vbraun/Code/sage/local/lib/libstdc++.so.6(+0x5dfa9)[0x7f06acab0fa9] > > /home/vbraun/Code/sage/local/lib/libpynac.so.0(py_error(char > > const*)+0x70)[0x7f069376ede0] > > > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::numeric::numeric(_object*, > > > > bool)+0x1b8)[0x7f069376f9e8] > > > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::numeric::add(GiNaC::numeric > > > > const&) const+0x1fa)[0x7f0693776c6a] > > > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::operator+(GiNaC::numeric > > > > const&, GiNaC::numeric const&)+0x9)[0x7f069377a3a9] > > > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::numeric::add_dyn(GiNaC::numeric > > > > const&) const+0x39)[0x7f0693770889] > > > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::expairseq::combine_overall_coeff(GiNaC::ex > > > > const&)+0x1a)[0x7f06936d163a] > > > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::expairseq::construct_from_2_ex(GiNaC::ex > > > > const&, GiNaC::ex const&)+0x375)[0x7f06936d6515] > > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::add::add(GiNaC::ex > > const&, GiNaC::ex const&)+0x63)[0x7f06936a5a73] > > > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::operator-(GiNaC::ex > > const&, GiNaC::ex const&)+0x5f)[0x7f069377b0af] > > > /home/vbraun/Code/sage/local/lib/libpynac.so.0(GiNaC::relational::decide() > > const+0x46)[0x7f069378b886] > > > /home/vbraun/Code/sage/local/lib/python2.7/site-packages/sage/symbolic/expression.so(+0x9ec48)[0x7f0693387c48] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyObject_IsTrue+0x37)[0x7f06c8dbaa57] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(+0x54e4e)[0x7f06c8d6fe4e] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(+0xbcb63)[0x7f06c8dd7b63] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyObject_Call+0x43)[0x7f06c8d6dc43] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x3b2e)[0x7f06c8e2228e] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCode+0x19)[0x7f06c8e256c9] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x58de)[0x7f06c8e2403e] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5936)[0x7f06c8e24096] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x830)[0x7f06c8e255a0] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyEval_EvalCode+0x19)[0x7f06c8e256c9] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyRun_FileExFlags+0x8a)[0x7f06c8e4949a] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(PyRun_SimpleFileExFlags+0xd7)[0x7f06c8e4aa27] > > > > > /home/vbraun/Code/sage/local/lib/libpython2.7.so.1.0(Py_Main+0xc25)[0x7f06c8e60c25] > > > > /lib64/libc.so.6(__libc_start_main+0xf0)[0x7f06c8054580] > > python(_start+0x29)[0x400769] > > > > > > > > > > > > On Sunday, January 17, 2016 at 5:39:43 PM UTC, Emmanuel Charpentier > wrote: > >> > >> (Note : Question also asked on ask.sagemath.org, but crossposted > because I > >> found a way to CRASH sage...). > >> I'm trying to understand coercions, and I'm hitting (repeatedly) > something > >> that I do not understand. > >> > >> Let's try to find thge roots of a polynom. We can try equation solving > (of > >> a quartic, no less) : > >> > >> sage: w = x^4 - (1+3*i)*x^3 - (2-4*i)*x^2 + (6-2*i)*x - 4 - 4*i > >> sage: S1=[t.rhs() for t in solve(w,x)];S1 > >> [-1/2*sqrt(2*I) + 3/2*I - 1/2, 1/2*sqrt(2*I) + 3/2*I - 1/2, -I + 1, I + > 1] > >> sage: bool(sqrt(2*I)==1+I) > >> True > >> > >> Or we can try the roots of a polynomial : > >> > >> sage: S2=[SR(t[0]) for t in w.roots(ring=QQbar)];S2 > >> [-1 + 1*I, 2*I, 1 - 1*I, 1 + 1*I] > >> sage: bool(sqrt(2*I)==1+I) > >> True > >> sage: S1R=[t.subs({sqrt(2*I):1+I}) for t in S1];S1R > >> [I - 1, 2*I, -I + 1, I + 1] > >> > >> So far, so good. I'm convinced that these two solution lists are one > and > >> the same. But I can't find a way to convince Sage: > >> > >> sage: map(lambda t,u:t-u, S1R, S2) > >> > >> > --------------------------------------------------------------------------- > >> TypeError Traceback (most recent call > >> last) > >> <ipython-input-6-be2e4183b522> in <module>() > >> ----> 1 map(lambda t,u:t-u, S1R, S2) > >> > >> <ipython-input-6-be2e4183b522> in <lambda>(t, u) > >> ----> 1 map(lambda t,u:t-u, S1R, S2) > >> > >> /usr/local/sage-7.0/src/sage/structure/element.pyx in > >> sage.structure.element.RingElement.__sub__ > >> > (/usr/local/sage-7.0/src/build/cythonized/sage/structure/element.c:15995)() > >> 1665 cdef long n > >> 1666 if have_same_parent_c(left, right): > >> -> 1667 return > >> (<ModuleElement>left)._sub_(<ModuleElement>right) > >> 1668 if PyInt_CheckExact(right): > >> 1669 n = PyInt_AS_LONG(right) > >> > >> /usr/local/sage-7.0/src/sage/symbolic/expression.pyx in > >> sage.symbolic.expression.Expression._sub_ > >> > (/usr/local/sage-7.0/src/build/cythonized/sage/symbolic/expression.cpp:20844)() > > > >> 2950 relational_operator(_right._gobj)) > >> 2951 else: > >> -> 2952 x = gsub(left._gobj, _right._gobj) > >> 2953 return new_Expression_from_GEx(left._parent, x) > >> 2954 > >> > >> /usr/local/sage-7.0/src/sage/structure/element.pyx in > >> sage.structure.element.RingElement.__add__ > >> > (/usr/local/sage-7.0/src/build/cythonized/sage/structure/element.c:15852)() > >> 1649 elif PyInt_CheckExact(left): > >> 1650 return > >> (<RingElement>right)._add_long(PyInt_AS_LONG(left)) > >> -> 1651 return coercion_model.bin_op(left, right, add) > >> 1652 > >> 1653 cdef RingElement _add_long(self, long n): > >> > >> /usr/local/sage-7.0/src/sage/structure/coerce.pyx in > >> sage.structure.coerce.CoercionModel_cache_maps.bin_op > >> > (/usr/local/sage-7.0/src/build/cythonized/sage/structure/coerce.c:9736)() > >> 1067 # We should really include the underlying error. > >> 1068 # This causes so much headache. > >> -> 1069 raise TypeError(arith_error_message(x,y,op)) > >> 1070 > >> 1071 cpdef canonical_coercion(self, x, y): > >> > >> TypeError: unsupported operand parent(s) for '+': 'Number Field in I > with > >> defining polynomial x^2 + 1' and 'Algebraic Field' > >> > >> I do not understand this error : both S1R and S2 are composed of things > >> belonging to SR : > >> > >> sage: map(lambda t:type(t), S1R) > >> [<type 'sage.symbolic.expression.Expression'>, > >> <type 'sage.symbolic.expression.Expression'>, > >> <type 'sage.symbolic.expression.Expression'>, > >> <type 'sage.symbolic.expression.Expression'>] > >> sage: map(lambda t:type(t), S2) > >> [<type 'sage.symbolic.expression.Expression'>, > >> <type 'sage.symbolic.expression.Expression'>, > >> <type 'sage.symbolic.expression.Expression'>, > >> <type 'sage.symbolic.expression.Expression'>] > >> > >> The reverse conversion works, however : > >> > >> sage: map(lambda t,u:t-u, [QQbar(t) for t in S1R], [QQbar(t) for t in > S2]) > >> [0, 0, 0, 0] > >> > >> Worse : testing directly for equality CRASHES Sage : > >> > >> sage: map(lambda t,u:bool(t==u), S1R, S2) > >> terminate called after throwing an instance of 'std::runtime_error' > >> what(): > >> > ------------------------------------------------------------------------ > >> > >> > /usr/local/sage-7.0/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x3c35)[0x7f3d47047c35] > > > >> > >> > /usr/local/sage-7.0/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x3c87)[0x7f3d47047c87] > > > >> > >> > /usr/local/sage-7.0/local/lib/python2.7/site-packages/sage/ext/interrupt/interrupt.so(+0x608c)[0x7f3d4704a08c] > > > >> /lib/x86_64-linux-gnu/libpthread.so.0(+0x10670)[0x7f3d4edc4670] > >> /lib/x86_64-linux-gnu/libc.so.6(gsignal+0x37)[0x7f3d4e337657] > >> /lib/x86_64-linux-gnu/libc.so.6(abort+0x16a)[0x7f3d4e338a2a] > >> > >> > /usr/lib/x86_64-linux-gnu/libstdc++.so.6(_ZN9__gnu_cxx27__verbose_terminate_handlerEv+0x15d)[0x7f3d3a9ad35d] > > > >> /usr/lib/x86_64-linux-gnu/libstdc++.so.6(+0x8d3b6)[0x7f3d3a9ab3b6] > >> /usr/lib/x86_64-linux-gnu/libstdc++.so.6(+0x8d401)[0x7f3d3a9ab401] > >> /usr/lib/x86_64-linux-gnu/libstdc++.so.6(+0x8d619)[0x7f3d3a9ab619] > >> > >> > /usr/local/sage-7.0/local/lib/libpynac.so.2(_Z8py_errorPKc+0x49)[0x7f3d214029c9] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaC7numericC1EP7_objectb+0x1b8)[0x7f3d21403508] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZNK5GiNaC7numeric3addERKS0_+0x1c2)[0x7f3d214094f2] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaCplERKNS_7numericES2_+0x9)[0x7f3d2140c569] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZNK5GiNaC7numeric7add_dynERKS0_+0x2d)[0x7f3d2140405d] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaC9expairseq21combine_overall_coeffERKNS_2exE+0x15)[0x7f3d2136f945] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaC9expairseq19construct_from_2_exERKNS_2exES3_+0x27d)[0x7f3d21374bed] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaC3addC1ERKNS_2exES3_+0x63)[0x7f3d21346633] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZN5GiNaCmiERKNS_2exES2_+0x5f)[0x7f3d2140d14f] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpynac.so.2(_ZNK5GiNaC10relational6decideEv+0x3e)[0x7f3d2141bede] > > > >> > >> > /usr/local/sage-7.0/local/lib/python2.7/site-packages/sage/symbolic/expression.so(+0xa7108)[0x7f3d21039108] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyObject_IsTrue+0x37)[0x7f3d4f06fe17] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(+0x54eae)[0x7f3d4f025eae] > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(+0xbc623)[0x7f3d4f08d623] > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyObject_Call+0x43)[0x7f3d4f023a73] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x3a6e)[0x7f3d4f0d634e] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(+0x835ac)[0x7f3d4f0545ac] > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyObject_Call+0x43)[0x7f3d4f023a73] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_CallObjectWithKeywords+0x47)[0x7f3d4f0d22e7] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(+0xfd564)[0x7f3d4f0ce564] > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5c8e)[0x7f3d4f0d856e] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCode+0x19)[0x7f3d4f0d93a9] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5234)[0x7f3d4f0d7b14] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x5a42)[0x7f3d4f0d8322] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7f3d4f0d928c] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyEval_EvalCode+0x19)[0x7f3d4f0d93a9] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyRun_FileExFlags+0x8a)[0x7f3d4f0fca9a] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(PyRun_SimpleFileExFlags+0xd7)[0x7f3d4f0fde47] > > > >> > >> > /usr/local/sage-7.0/local/lib/libpython2.7.so.1.0(Py_Main+0xc3e)[0x7f3d4f1143ee] > > > >> /lib/x86_64-linux-gnu/libc.so.6(__libc_start_main+0xf0)[0x7f3d4e324870] > >> python(_start+0x29)[0x4006f9] > >> > ------------------------------------------------------------------------ > >> Attaching gdb to process id 11241. > >> > >> Saved trace to /home/charpent/.sage/crash_logs/sage_crash_qQFZhv.log > >> > ------------------------------------------------------------------------ > >> Unhandled SIGABRT: An abort() occurred in Sage. > >> This probably occurred because a *compiled* component of Sage has a bug > >> in it and is not properly wrapped with sig_on(), sig_off(). > >> Sage will now terminate. > >> > ------------------------------------------------------------------------ > >> Abandon > >> > >> > > -- > > You received this message because you are subscribed to the Google > Groups > > "sage-support" group. > > To unsubscribe from this group and stop receiving emails from it, send > an > > email to sage-support...@googlegroups.com <javascript:>. > > To post to this group, send email to sage-s...@googlegroups.com > <javascript:>. > > Visit this group at https://groups.google.com/group/sage-support. > > For more options, visit https://groups.google.com/d/optout. > > > > -- > William (http://wstein.org) > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at https://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
