On May 14, Daniel Loughran wrote:
I don't follow you; given a rational point on an elliptic curve you can always take its reduction modulo any integer.The equation of the projective curve is: Y^2*Z + Y*Z^2 = X^3 - X*Z^2 When you plug in (X:Y:Z) = (2:1:0), all terms involving Z vanish, so you are left with the equation 0 = 2^3 modulo 4,
My bad, I didn't even check it, just tried: sage: E4(2,1,0) Traceback: ... TypeError: Coordinates [2, 1, 0] do not define a point on Elliptic Curve defined by y^2 + y = x^3 + 3*x over Ring of integers modulo 4
which indeed holds. On Thursday, 14 May 2020 13:29:32 UTC+1, Reimundo Heluani wrote: On May 14, 'Reimundo Heluani' via sage-devel wrote: >On May 14, Daniel Loughran wrote: >>Hello. I think that I may have found a bug involving elliptic curves modulo >>powers of primes. I have attached the working jupyter notebook, but my code and >>results are also below. >> >>In my code I have an elliptic curve E over Q with good reduction at 2 and the >>point P = (2:-3:8) (homogeneous coordinates). >> >>It is clear that the reduction modulo 4 of this point in projective space is >>(2:1:0). >> >[2:1:0] is not in that curve as > >1^2*0 + 1*0^2 != 1^3 - 1*0^2 > OOops, the LHS is 2^3 - 2*0^2 but still not equal to the RHS :) R. > >The reduction given by sage is however > >R >> >>However, sage is telling me that when I reduce the curve modulo 4 then ask what >>point P reduces to, I get the identity element (0:1:0). >> >>My guess is that sage does something like put this point into the form (1 /4:-3/ >>8:1), then notices that 4 divides the denominator of the x-coordinate so just >>kills it. But really it should be clearing denominators before it tries to >>reduce modulo 4. >> >>There is another related bug: if I try to instead reduce this point modulo 16, >>then I just get the error "inverse of Mod(4, 16) does not exist". I guess this >>is a similar problem to the above. >> >> >>---------------------------------------------------------- -------------------------------------------------------------------- >>[ ] >> >>E=EllipticCurve([0, 0, 1, -1, 0]); E >> >>Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field >> >> >>[ ] >> >>P=E(2,-3,8); P >> >>(1/4 : -3/8 : 1) >> >> >>[ ] >> >>E4=E.change_ring(Integers(4)); E4 >> >>Elliptic Curve defined by y^2 + y = x^3 + 3*x over Ring of integers modulo 4 >> >> >>[ ] >> >>E4(P) >> >>(0 : 1 : 0) >> >> >>[ ] >> >>P2.<X,Y,Z> = ProjectiveSpace(Integers(4),2); P2 >> >>Projective Space of dimension 2 over Ring of integers modulo 4 >> >> >>[ ] >> >>P2(2,-3,8) >> >>(2 : 1 : 0) >> >> >>[ ] >> >>P2(0,1,0) >> >>(0 : 1 : 0) >> >> >>[ ] >> >>P2(2,-3,8)==P2(0,1,0) >> >>False >> >> >>[ ] >> >>E16=E.change_ring(Integers(16)); E16 >> >>Elliptic Curve defined by y^2 + y = x^3 + 15*x over Ring of integers modulo 16 >> >> >>[ ] >> >>E16(P) >>ZeroDivisionError: inverse of Mod(4, 16) does not exist >> >>-- >>You received this message because you are subscribed to the Google Groups >>"sage-devel" group. >>To unsubscribe from this group and stop receiving emails from it, send an email >>to [11][1]sage-...@googlegroups.com. >>To view this discussion on the web visit [12][2]https://groups.google.com /d/msgid/ >>sage-devel/f2900af4-6aad-4450-9e12-6f1ec95596f8%[3]40googlegroups.com. >> >>References: >> >>[11] mailto:[4]sage-...@googlegroups.com >>[12] [5]https://groups.google.com/d/msgid/sage-devel/f2900af4- 6aad-4450-9e12-6f1ec95596f8%40googlegroups.com?utm_medium=email&utm_source= footer > > >-- >You received this message because you are subscribed to the Google Groups "sage-devel" group. >To unsubscribe from this group and stop receiving emails from it, send an email to [6]sage-...@googlegroups.com. >To view this discussion on the web visit [7]https://groups.google.com/d/ msgid/sage-devel/20200514121620.GA39760%40vertex. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to [8]sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit [9]https://groups.google.com/d/msgid/ sage-devel/344c58f2-a339-43c6-8417-93d46c05d0f7%40googlegroups.com. References: [1] javascript: [2] https://groups.google.com/d/msgid/ [3] http://40googlegroups.com/ [4] javascript: [5] https://groups.google.com/d/msgid/sage-devel/f2900af4-6aad-4450-9e12-6f1ec95596f8%40googlegroups.com?utm_medium=email&utm_source=footer [6] javascript: [7] https://groups.google.com/d/msgid/sage-devel/20200514121620.GA39760%40vertex [8] mailto:sage-devel+unsubscr...@googlegroups.com [9] https://groups.google.com/d/msgid/sage-devel/344c58f2-a339-43c6-8417-93d46c05d0f7%40googlegroups.com?utm_medium=email&utm_source=footer
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