2024-02-11 20:06:07 +0200 received badge ● Notable Question (source) 2023-10-07 05:56:04 +0200 received badge ● Popular Question (source) 2023-10-07 05:56:04 +0200 received badge ● Notable Question (source) 2023-07-12 18:09:24 +0200 received badge ● Notable Question (source) 2023-06-29 08:41:28 +0200 received badge ● Notable Question (source) 2023-06-19 18:16:30 +0200 received badge ● Popular Question (source) 2023-06-19 18:16:30 +0200 received badge ● Notable Question (source) 2023-05-09 23:04:05 +0200 received badge ● Famous Question (source) 2023-02-27 14:42:35 +0200 received badge ● Notable Question (source) 2023-01-23 18:31:15 +0200 received badge ● Famous Question (source) 2022-10-22 10:44:49 +0200 received badge ● Popular Question (source) 2022-10-04 00:44:40 +0200 marked best answer finding 4-torsion point on elliptic curve Hi I have the following: E=EllipticCurve(Qp,[0,0,0,-3267,45630]) P=E([-21,324])  Is there some coding I can use to get the 4-torsion point of this elliptic curve. I reckon it might have something to do with the nth divisional polynomial but I couldn't find it. 2022-10-04 00:43:04 +0200 received badge ● Famous Question (source) 2022-10-02 22:22:50 +0200 received badge ● Popular Question (source) 2022-05-18 14:49:30 +0200 received badge ● Famous Question (source) 2021-11-21 17:57:53 +0200 received badge ● Popular Question (source) 2021-08-13 00:10:52 +0200 received badge ● Famous Question (source) 2021-05-18 21:09:55 +0200 received badge ● Popular Question (source) 2021-05-05 03:58:02 +0200 received badge ● Notable Question (source) 2021-03-14 12:19:02 +0200 received badge ● Famous Question (source) 2021-03-07 21:37:33 +0200 received badge ● Famous Question (source) 2021-02-09 12:43:12 +0200 received badge ● Popular Question (source) 2021-02-08 19:50:54 +0200 received badge ● Popular Question (source) 2021-02-08 19:50:40 +0200 received badge ● Notable Question (source) 2021-01-27 19:53:40 +0200 received badge ● Popular Question (source) 2021-01-27 19:18:43 +0200 received badge ● Popular Question (source) 2021-01-27 19:18:43 +0200 received badge ● Notable Question (source) 2020-12-14 20:47:33 +0200 received badge ● Notable Question (source) 2020-11-08 11:15:04 +0200 received badge ● Popular Question (source) 2020-11-07 06:25:37 +0200 received badge ● Notable Question (source) 2020-11-07 06:25:37 +0200 received badge ● Popular Question (source) 2020-10-05 22:44:53 +0200 received badge ● Notable Question (source) 2020-10-05 22:44:53 +0200 received badge ● Popular Question (source) 2020-04-04 22:21:19 +0200 received badge ● Popular Question (source) 2020-03-20 23:04:16 +0200 received badge ● Popular Question (source) 2020-02-26 14:02:41 +0200 commented question for-loop not computing the values @johnpalmieri thank you for the feedback John. I really appreciate it. 2020-02-26 14:01:46 +0200 commented question for-loop not computing the values @dsejas Hi diego. thank you for your reply. I am actually trying to figure out the formula too.. just to check if there's anything wrong with it. I don't have a deadline with this. Thank you for your help. Let me too re check the calculation manually. 2020-02-25 09:53:13 +0200 asked a question for-loop not computing the values Hi. I am quite new to using phyton is SAGE, but what I did here is that I have tried my best to come up with the code based on reading multiple notes on SAGE. The values of alpha, n, eta, lambda, h, T, t, kappa, k(x-s) and gamma values are given. I have tried to do the coding from scratch and this is what I have obtained: from __future__ import print_function alpha = 1.2 n = 2 eta = 1.3 lamda = 1 h = 1 T = 1 t = 0.1 h, t, x, s = var('h t x s') U[t] = 1/3*t^3+1 P[t] = 2*t-lamda*(-t^4/24+t^5/15-t^7/126+t^8/72) b[k] = 1 def X(m): if m <= 1: return 0 else: return 1 k[x,s]=s-x gamma[z]=integrate(x^(z-1)*e^(-x),x, 0, infinity) u[0]=1 for m in range(1,4): u[m] = sum(b[k]/factorial(k)*t^k,k,0,n-1) + X[m]*u[m-1]-X[m]*sum(𝚍𝚎𝚛𝚒𝚟𝚊𝚝𝚒𝚟𝚎(u[m-1][0],t,k)/factorial(k)*t^k,k,0,n-1)+h/gamma[eta-alpha]*integrate((t-tau)^(eta-alpha-1)*u[m-1].subs(t=tau)*P[t].subs(t=tau),tau, 0, t); print(u[m].full_simplify()) UU=u[1]+u[2]+u[3]+u[4]:  What I am trying to achieve here is the values of u_1, u_2, u_3, u_4. Then calculate UU = u_1 + u_2 + u_3 + u_4. At the moment, I am not getting the answer. Not sure what is causing the error. It took me so much time and effort to come up with this from scratch, hope someone can help me indicate what's wrong with this coding. Thank you in advance. 2020-02-17 04:33:07 +0200 commented question looping of equality function oh sorry... it was giving an error code every time I ran the code. 2020-02-13 09:53:18 +0200 commented question looping of equality function the answers I am planning to obtain is the list of numbers for x, y, z that satisfies the equation x^4+y^4=z^3. 2020-02-13 04:54:59 +0200 marked best answer programming of looping to print selected value of m I have the following code for m=1: Qp=pAdicField(13) E=EllipticCurve(Qp,[0,0,0,-3267,45630]); E P=E([-21,324]);P p=13;p #because I am working 13-adically R=12104*1*P;R #m=1 S=-2*P;S Q=R+S;Q (12 + 8*13 + 5*13^2 + 10*13^3 + 10*13^4 + O(13^5) : 9 + 3*13 + 6*13^2 + 13^3 + 5*13^4 + 2*13^5 + O(13^5) : 1 + O(13^5)) x=Q[0];x 12 + 8*13 + 5*13^2 + 10*13^3 + 10*13^4 + O(13^5) y=Q[1];y 9 + 3*13 + 6*13^2 + 13^3 + 5*13^4 + O(13^5) W=(81*x^16 + 40662*x^15 + 14353281*x^14 - 460241028*x^13 - 644722959186*x^12 + 39379675354740*x^11 + 5212980804862026*x^10 - 415546630058854656*x^9 - 8202010485984353739*x^8 + 1396767997483732402758*x^7 - 27550698906220673513787*x^6 - 1044392234943529703379852*x^5 + 60770398462922893831446348*x^4 - 1284453663719469166478575296*x^3 + 14183844641879715988450074288*x^2 - 81800517874945025246941522368*x + 196162341839727571433321441856)- (3240*x^14 + 456840*x^13 + 188268624*x^12 - 45834271200*x^11 - 2435651997264*x^10 + 682353767281968*x^9 - 7053953405575680*x^8 - 2553415737499629216*x^7 + 98906717445152189544*x^6 + 1348117411901578667784*x^5 - 162666175355778441465360*x^4 + 4276857451171442758058304*x^3 - 54456600108308451946891776*x^2 + 350065581968511893813480064*x - 918312303919436410092339456)*y;W 2 + 4*13 + 3*13^2 + 11*13^3 + 11*13^4 + O(13^5) D=W[0];D 2 T=kronecker(D,p);T -1  So in this example I reject m=1 because T=-1. So basically I want to do a loop for m=1,2,...,10, and everytime T=-1 I will reject that value of m and only leave set of numbers of m=[2,..10] that gives T=1. I tried doing the coding to work but it keeps giving error regarding the "m" value. I did something like this Qp=pAdicField(13) E=EllipticCurve(Qp,[0,0,0,-3267,45630]); E P=E([-21,324]);P p=13;p #because I am working 13-adically R=12104*m*P;R #I use a general m which I will define later that m=1,2,...,10 S=-2*P;S Q=R+S;Q x=Q[0];x y=Q[1];y W=(81*x^16 + 40662*x^15 + 14353281*x^14 - 460241028*x^13 - 644722959186*x^12 + 39379675354740*x^11 + 5212980804862026*x^10 - 415546630058854656*x^9 - 8202010485984353739*x^8 + 1396767997483732402758*x^7 - 27550698906220673513787*x^6 - 1044392234943529703379852*x^5 + 60770398462922893831446348*x^4 - 1284453663719469166478575296*x^3 + 14183844641879715988450074288*x^2 - 81800517874945025246941522368*x + 196162341839727571433321441856)- (3240*x^14 + 456840*x^13 + 188268624*x^12 - 45834271200*x^11 - 2435651997264*x^10 + 682353767281968*x^9 - 7053953405575680*x^8 - 2553415737499629216*x^7 + 98906717445152189544*x^6 + 1348117411901578667784*x^5 - 162666175355778441465360*x^4 + 4276857451171442758058304*x^3 - 54456600108308451946891776*x^2 + 350065581968511893813480064*x - 918312303919436410092339456)*y;W D=W[0];D T=kronecker(D,p);T for m in range(10) : if T == 1: print(m)  Unfortunately, the programming is not working, keeps giving error for m. By the way, I got the ... 2020-02-13 04:54:27 +0200 commented answer Problem with integrating the expression of M thank you! this is great! 2020-02-13 04:54:13 +0200 marked best answer Problem with integrating the expression of M Hi. I have the following code that I want to integrate and differentiate, but I am stuck at the expression of M, where it is unable to integrate. Is my coding wrong? c,t = var('c t') Pi = RR.pi() G=integrate(sqrt(1-t^2)*(t+c),t,-0.9,0.9);G H=G.diff(c);H L=integrate(-(t+c)/(sqrt(1-t^2)),t,-0.9,0.9);L M=integrate(1/(sqrt(1-t^2)*(t-c)),t,-0.9,0.9);M #cannot seem to integrate this wrt to t I=(c^2-1)*(L+(1-c^2)*M);I #equation I that involves M P=I.diff(c);P #differentiate I wrt to c to obtain equation P  I have tried integrating M by hand which gives me a closed-form involving log function, but I can't seem to integrate it here using Sage. 2020-02-13 04:52:34 +0200 asked a question looping of equality function Hi, I have the equation x^4+y^4=z^3. I want to run each variable from -100 till 100 to see which numbers satisfy this equation. Here is what I have done: x,y,z= var('x y z') for x in range(-100, 100): for y in range(-100, 100): for z in range(-100, 100): x^4+y^4==z^3 print(x,y,z)  There is definitely something wrong with the coding that's not giving me the desired answer. Can someone enlighten me, please.