#23222 closed defect (fixed)
Add test to EllipticCurve.isogeny() to check for validity of kernel polynomials
Reported by:  defeo  Owned by:  

Priority:  major  Milestone:  sage8.7 
Component:  elliptic curves  Keywords:  isogenies, vélu's formulas 
Cc:  cremona  Merged in:  
Authors:  John Cremona  Reviewers:  Luca De Feo, Frédéric Chapoton 
Report Upstream:  N/A  Work issues:  
Branch:  28e369f (Commits)  Commit:  
Dependencies:  Stopgaps: 
Description (last modified by )
In this example
sage: K = GF(47^2) sage: E = EllipticCurve([0, K.gen()]) sage: f = E.division_polynomial(7).factor()[4][0] sage: E.is_supersingular() True sage: E.isogeny(f).codomain().is_supersingular() False sage: phi = E.isogeny(E.division_polynomial(7).factor()[8][0]) sage: phi(E.random_point())  TypeError Traceback (most recent call last) ... TypeError: Coordinates [16*z2 + 30, 3*z2 + 22, 1] do not define a point on Elliptic Curve defined by y^2 = x^3 + (3*z2+36)*x over Finite Field in z2 of size 47^2
The factors of the seventh division polynomial are not detected as bad kernel polynomials, and a wrong isogeny is computed instead.
Tests to detect these cases are in .isogenies_prime_degree_general()
. They should be moved to an independent function and called before applying Vélu's formulas.
Change History (31)
comment:1 followup: ↓ 2 Changed 2 years ago by
comment:2 in reply to: ↑ 1 ; followup: ↓ 5 Changed 2 years ago by
Replying to cremona:
Remember that the Trace of Frobenius is an integer, not an element of the field, so seeing 94 which is 0 mod 47 is ok.
I agree with that, but doesn't SatoTate say that two curves are isogenous over a finite field iff they have the same trace? As I said, j=0 and j=36 are isogenous over GF(47^{2}), but the exact curves in the example are not. And, indeed:
sage: phi = E.isogeny(E.division_polynomial(7).factor()[8][0]) sage: phi(E.random_point())  TypeError Traceback (most recent call last) ... TypeError: Coordinates [16*z2 + 30, 3*z2 + 22, 1] do not define a point on Elliptic Curve defined by y^2 = x^3 + (3*z2+36)*x over Finite Field in z2 of size 47^2
Secondly, you have omitted the isogeny whose kernel is the product of the 3 linear factors of the 7division polynomial.
Yes. I just wanted to highlight the bug. One of the buggy factors would have been enough.
Thirdly, if you want to see the 7isogenous curves it is better to do
E.isogenies_prime_degree(7)since it is nontrivial to combine the factors of the division polynomial correctly.
Absolutely. However, I found the bug while doing a more general search.
Not that E.isogeny(f) does *not* actually check that f is a valid kernel polynomial even when check=True is give (the default). It should! When it was written we did not have code to do that but now we do. It is slightly buried in the function isogenies_prime_degree_general().
Well, that's not completely correct:
sage: E.isogeny(E.base_field().polynomial_ring().random_element(3))  ValueError Traceback (most recent call last) ... ValueError: The polynomial does not define a finite subgroup of the elliptic curve.
So there are some checks, but they are buggy, at least for j=0,1728, I suspect.
I do not have time to dig this further right now. Hopefully in a couple of weeks I can come up with a fix.
comment:3 Changed 2 years ago by
The jinvariant does not determine the trace since it only determines the curve up to twist; in your case there are more than 2 twists as well.
Yes, some checking is done for validity of kernel polys, but that is misleading since bad input passes those tests as they are not complete.
We'll come back to that.
comment:4 Changed 17 months ago by
Can we close this as invalid / won't fix? I just reread the comments and think that I explained why none of the output which had been thought to be suspect was incorrect.
comment:5 in reply to: ↑ 2 Changed 17 months ago by
I still don't get it. Admittedly, E.isogeny(f)
does not test that f
is a valid kernel polynomial. But, why is f
not a valid kernel polynomial in this example:
sage: f = E.division_polynomial(7).factor()[4][0] sage: E.is_supersingular() True sage: E.isogeny(f).codomain().is_supersingular() False
Also, the TypeError
in comment 2 is not nice and would deserve fixing.
comment:6 Changed 17 months ago by
The kernel poly of a 7isogeny is a degree 3 factor of the 7division poly but not every degree 3 factor is a kernel poly! In this case the 3 roots of your degree 3 factor, which all lie in GF(47^{6), are the xcoordinates of points in 3 different subgroups of order 7. }
In the function isogenies_prime_degree_general() in sage.schemes.elliptic_curves.isogeny_small_degree, this is implemented properly: factor the ldivision poly (here l is an odd prime); keep only factors of degree dividing (l1)/2; for each such factor work out whether or not it is an irreducible factor of a kernel poly.
It would be possible to extract from the code of that function a test for whether a polynomial of degree (l1)/2 actually is a kernel poly. Feel free to make a ticket for that! The relevent theory is in Kimi Tsukazaki's thesis (cited in that file).
comment:7 Changed 17 months ago by
Oh, right!
Can't we just edit this ticket so that it points to the problem, instead of closing it and opening a new one?
comment:8 Changed 17 months ago by
Sure, go ahead. Once we have a standalone function test_kernel_poly(E,f,ell) it is clear where to put a call to it: in place of lines 24502454 where now it only tests if the polynomial divides the division poly. I should have done this ages ago.
comment:9 Changed 17 months ago by
 Description modified (diff)
 Keywords vélu's formulas added; supersingular removed
 Summary changed from Incorrect output from Vélu's formulas on j=0? to Add test to EllipticCurve.isogeny() to check for validity of kernel polynomials
 Type changed from defect to enhancement
comment:11 Changed 16 months ago by
 Branch set to u/cremona/23222
 Commit set to 0c120af3a0cb8323954bd5994984bc433adadfe8
 Milestone changed from sage8.0 to sage8.2
 Status changed from new to needs_review
New commits:
0c120af  #23222: check validity of isogeny kernel polynomials

comment:12 Changed 16 months ago by
 Commit changed from 0c120af3a0cb8323954bd5994984bc433adadfe8 to f7518f7da457d7dc25a32a0ba9e92ab58a189b45
Branch pushed to git repo; I updated commit sha1. New commits:
f7518f7  #23222: fix docstring formatting error

comment:13 followup: ↓ 14 Changed 9 months ago by
 Status changed from needs_review to needs_work
red banch, needs rebasing
comment:14 in reply to: ↑ 13 Changed 9 months ago by
Replying to chapoton:
red branch, needs rebasing
I will rebase it, but it would be good to have a commitment from someone to review it, since the actual work on this was done over 6 months ago.
comment:15 Changed 9 months ago by
 Commit changed from f7518f7da457d7dc25a32a0ba9e92ab58a189b45 to 0683290a5a04626d57eed1e28910bfc9346a6f34
Branch pushed to git repo; I updated commit sha1. New commits:
0683290  Merge branch 'develop' into 23222

comment:16 Changed 9 months ago by
 Status changed from needs_work to needs_review
comment:17 Changed 9 months ago by
It's been in my todo list for 6 months indeed. But I won't have time before Christmas.
comment:18 Changed 7 months ago by
I have one doctest failure:
sage t long warnlong 83.4 src/sage/schemes/elliptic_curves/ell_field.py ********************************************************************** File "src/sage/schemes/elliptic_curves/ell_field.py", line 1023, in sage.schemes.elliptic_curves.ell_field.EllipticCurve_field.isogeni$ s_prime_degree Failed example: E.isogenies_prime_degree(5) Exception raised: Traceback (most recent call last): File "/data/dfl/sage/local/lib/python2.7/sitepackages/sage/doctest/forker.py", line 671, in _run self.compile_and_execute(example, compiler, test.globs) File "/data/dfl/sage/local/lib/python2.7/sitepackages/sage/doctest/forker.py", line 1086, in compile_and_execute exec(compiled, globs) File "<doctest sage.schemes.elliptic_curves.ell_field.EllipticCurve_field.isogenies_prime_degree[28]>", line 1, in <module> E.isogenies_prime_degree(Integer(5)) File "/data/dfl/sage/local/lib/python2.7/sitepackages/sage/schemes/elliptic_curves/ell_field.py", line 1091, in isogenies_prime$ degree return sum([isogenies_prime_degree(self, d) for d in L], []) File "/data/dfl/sage/local/lib/python2.7/sitepackages/sage/schemes/elliptic_curves/isogeny_small_degree.py", line 2328, in isog$ nies_prime_degree return isogenies_prime_degree_general(E,l, minimal_models=minimal_models) File "/data/dfl/sage/local/lib/python2.7/sitepackages/sage/schemes/elliptic_curves/isogeny_small_degree.py", line 2197, in isog$ nies_prime_degree_general return [E.isogeny(k) for k in ker] File "/data/dfl/sage/local/lib/python2.7/sitepackages/sage/schemes/elliptic_curves/ell_field.py", line 880, in isogeny raise RuntimeError("Unable to construct isogeny: %s" % e) RuntimeError: Unable to construct isogeny: 'sage.rings.fraction_field_FpT.FpTElement' object has no attribute 'divides' **********************************************************************
...investigating.
comment:19 Changed 7 months ago by
The doctest failure is due to #27064. No issue to be addressed here.
comment:20 Changed 7 months ago by
In is_kernel_polynomial
, there is this test:
psi_m = E.division_polynomial(m) if not f.divides(psi_m): return False
isn't it (much) more efficient to do
if E.division_polynomial(m, x=f.parent().quo(f).gen()) != 0: return False
comment:21 Changed 7 months ago by
The doc still says
* "check" (default: True) does some partial checks that the input is valid (e.g., that the kernel polynomial provided is valid); however, invalid input can in some cases still pass, since that the points define a group is not checked.
what are the invalid inputs that still pass?
comment:22 Changed 7 months ago by
 Milestone changed from sage8.2 to sage8.7
 Status changed from needs_review to needs_work
I will correct the docstring and also avoid using f.divides as suggested.
comment:23 Changed 7 months ago by
 Commit changed from 0683290a5a04626d57eed1e28910bfc9346a6f34 to 72a61bbd024b6b089b71aa8a3405f07e6fb55ae2
Branch pushed to git repo; I updated commit sha1. New commits:
f123620  Merge branch 'develop' of trac.sagemath.org:sage into 23222

0c30e53  Merge branch 'develop' into 23222

7e7593f  #23222 small change to avoid #27064, and update docstring

72a61bb  Merge remotetracking branch 'trac/u/cremona/23222' into 23222

comment:24 Changed 7 months ago by
 Reviewers set to defeo, chapoton
 Status changed from needs_work to needs_review
comment:25 Changed 7 months ago by
 Branch changed from u/cremona/23222 to u/defeo/23222
comment:26 Changed 7 months ago by
 Commit changed from 72a61bbd024b6b089b71aa8a3405f07e6fb55ae2 to 28e369f252fa8339ba379c5619096016afdc4493
Branch pushed to git repo; I updated commit sha1. New commits:
f123620  Merge branch 'develop' of trac.sagemath.org:sage into 23222

0c30e53  Merge branch 'develop' into 23222

7e7593f  #23222 small change to avoid #27064, and update docstring

72a61bb  Merge remotetracking branch 'trac/u/cremona/23222' into 23222

28e369f  Merge branch 'u/cremona/23222' of git://trac.sagemath.org/sage into t/23222/23222

comment:27 Changed 7 months ago by
 Reviewers changed from defeo, chapoton to Luca De Feo, Fréderic Chapoton
 Status changed from needs_review to positive_review
Doctests pass. I just fixed a minor problem in the docstring (r"""
instead of """
). Thanks, and sorry for the long time.
comment:28 Changed 7 months ago by
Thanks!
comment:29 Changed 7 months ago by
 Reviewers changed from Luca De Feo, Fréderic Chapoton to Luca De Feo, Frédéric Chapoton
comment:30 Changed 7 months ago by
 Branch changed from u/defeo/23222 to 28e369f252fa8339ba379c5619096016afdc4493
 Resolution set to fixed
 Status changed from positive_review to closed
comment:31 Changed 7 months ago by
 Commit 28e369f252fa8339ba379c5619096016afdc4493 deleted
one py3incompatible import was added here ; fix at #27107
Remember that the Trace of Frobenius is an integer, not an element of the field, so seeing 94 which is 0 mod 47 is ok. Secondly, you have omitted the isogeny whose kernel is the product of the 3 linear factors of the 7division polynomial. Thirdly, if you want to see the 7isogenous curves it is better to do
since it is nontrivial to combine the factors of the division polynomial correctly. In this case there are only two 7isogenies. Not that E.isogeny(f) does *not* actually check that f is a valid kernel polynomial even when check=True is give (the default). It should! When it was written we did not have code to do that but now we do. It is slightly buried in the function isogenies_prime_degree_general().