Is there a faster way (< 1 hour) to compute the rank of the following curve?
sage: E=EllipticCurve([525,228,0,-14972955,856475^2])
sage: E.two_descent()
Working with minimal curve [1,-1,1,-1608154463,25555312501831] via [u,r,s,t] = [1,-23045,-262,6049313]
Basic pair: I=77191414209, J=-44158885280436546
disc=-110222528210564668480677268684800
2-adic index bound = 2
By Lemma 5.1(b), 2-adic index = 1
2-adic index = 1
One (I,J) pair
*** BSD give two (I,J) pairs
Looking for quartics with I = 77191414209, J = -44158885280436546
Looking for Type 3 quartics:
Trying positive a from 1 up to 97338 (square a first...)
(1,0,-2569108581,70882136446950,-550026568648034946) --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [3425478110:70880423707891:8]
height = 19.3580374139982139686097624130447703630897684162794
Rank of B=im(eps) increases to 1
(1,0,-79008759,382273295436,-520192233942156) --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [105345014:382220622925:8]
height = 16.6812813105166386389509418874147230151015412188468
Rank of B=im(eps) increases to 2
(1,0,-164229,11437514,4185020814) --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [218974:11328023:8]
height = 11.4827447652308852790130227025168290728280040230824
Rank of B=im(eps) increases to 3
(1,0,106209,55699200,5492588544) --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [-141610:55770001:8]
height = 10.3302332544875895056301104557603387856715464732049
Rank of B=im(eps) increases to 4
(1,-1,-154173,8834360,4449633100) --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [1644532:69235889:64]
height = 11.7002057333692153746721973178788430852437315582425
Rank of B=im(eps) increases to 5
(1,2,-50825775,197184311888,-215166592255424) --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [8470963:24650156726:1]
height = 12.7212304727049559592490676880048355572479951461661
Rank of B=im(eps) increases to 6
(1,2,-457587,148456132,-10941975964) --nontrivial...--new (A) #1 (x:y:z) = (1 : 1 : 0)
Point = [76265:18576082:1]
height = 9.80847862275472614146303766858310619081781113771917
Size of A=ker(eps) increases to 2
(1,2,275793,12561872,100417216) --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [-45965:1558742:1]
height = 7.1124160555385017138543826219232258062506120422326
Rank of B=im(eps) increases to 7
(9,-6,-457587,49790436,-1232321076) --nontrivial...--equivalent to (A) #1
(36,-47,-115377,1833888,147270796) --nontrivial...--new (A) #2 (x:y:z) = (1 : 6 : 0)
Point = [265885080:18099825973:13824]
height = 15.5988971231435288000655531265862695040572334387229
Size of A=ker(eps) increases to 3
(81,-18,-457587,16596812,-136924564) --nontrivial...--equivalent to (A) #1
(81,90,-457551,16291760,-131443136) --nontrivial...--equivalent to (A) #1
(81,-126,-457515,16901852,-142507676) --nontrivial...--equivalent to (A) #1
(289,390,-3583851,214362260,-3608994444) --nontrivial...--equivalent to (A) #2
(324,-573,-115020,688119,16146846) --nontrivial...--equivalent to (A) #2
(361,-142,224037,2394828,5996916) --nontrivial...--new (A) #3 (x:y:z) = (1 : 19 : 0)
Point = [-256085971:39857828834:6859]
height = 15.7843641858060237367375581970740197608170491286891
Size of A=ker(eps) increases to 4
(361,-263,-314667,4008872,-5767984) --nontrivial...--equivalent to (A) #1
(529,578,21429,1954188,12621420) --nontrivial...--equivalent to (A) #3
(625,278,-146691,262908,7452324) --nontrivial...--equivalent to (A) #3
(1849,-3530,-40851,744300,3048516) --nontrivial...--equivalent to (A) #2
(2916,-1719,-115020,229373,1794094) --nontrivial...--equivalent to (A) #2
(2916,2169,-114795,152552,1857744) --nontrivial...--equivalent to (A) #2
(2916,-5607,-111357,305048,1704956) --nontrivial...--equivalent to (A) #2
(3249,-426,224037,798276,666324) --nontrivial...--equivalent to (A) #3
(3249,3543,-313290,1126731,-230409) --nontrivial...--equivalent to (A) #1
(4761,1734,21429,651396,1402380) --nontrivial...--equivalent to (A) #3
(5625,834,-146691,87636,828036) --nontrivial...--equivalent to (A) #3
(7569,8002,-104871,86952,751760) --nontrivial...--equivalent to (A) #3
(9025,6510,225777,568784,344640) --nontrivial...--equivalent to (A) #3
(9409,3714,-108447,104576,589824) --nontrivial...--equivalent to (A) #2
(13225,-18270,30657,371696,352104) --nontrivial...--equivalent to (A) #3
(14641,-8350,-144351,101920,306224) --nontrivial...--equivalent to (A) #1
(15625,26390,-130023,-60104,296144) --nontrivial...--equivalent to (A) #3
(16641,-10590,-40851,248100,338724) --nontrivial...--equivalent to (A) #2
