Rational points on elliptic curves over the rationals
ellrank
? E = ellinit([0,-1,0,-260,-1530]);
? ellrank(E)
%8 = [1,3,[[27,102]]]
Here the rank is either 1 or 3 and one point is known. Here the
conductor is small so we can check the analytic rank:
? A=ellanalyticrank(E)
%9 = [1,4.2585990440444049230727399816153311674]
We find that the analytic rank is 1 so the rank is 1 and we have
a Q-basis, and ØE is even
? A[2]/ellbsd(E)/ellheight(E,[27,102])
%10 = 4.0000000000000000000000000000000000000
So we can conclude under BSD that |Ø(E)| = 4