CoCalc Public Fileswww / 168 / notes / 2005-10-17 / 2005-10-17-sage-log.txt
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1[email protected]:~$cd 168/notes/2005-10-17 2[email protected]:~/168/notes/2005-10-17$ sage
3------------------------------------------------------------------------
4  SAGE Version 0.7.13, Export Date: 2005-10-17-0407
6  IPython shell -- for help type <object>?, <object>??, %magic, or help
7------------------------------------------------------------------------
8
9sage: (3/2)^2 + (20/3)^2
10 _1 = 1681/36
11sage: (41/6)^2
12 _2 = 1681/36
13sage: attach "cong.sage"
14sage: E = congru
15congruent_curve_gens  congruent_triangle
16sage: E = cong_num
17cong_number_curve  cong_number_sets
18sage: E = cong_number_curve(5)
19sage: E
20 _5 = Elliptic Curve defined by y^2  = x^3 - 25*x over Rational Field
21sage: G = E.gens()
22sage: G
23 _7 = [(-4, 6)]
24sage: P = G[0]
25sage: P
26 _9 = (-4, 6)
27sage: congrue
28congruent_curve_gens  congruent_triangle
29sage: congruent_triangle(P)
30_10 = (3/2, 20/3, 41/6)
31sage: 2*P
32_11 = (1681/144, -62279/1728)
33sage: congruent_triangle(2*P)
34_12 = (1519/492, 4920/1519, -3344161/747348)
35sage: a,b,c = _12
36sage: a
37_14 = 1519/492
38sage: b
39_15 = 4920/1519
40sage: c
41_16 = -3344161/747348
42sage: a^2 + b^2 == c^2
43_17 = True
44sage: a*b/2
45_18 = 5
46sage: a,b,c
47_19 = (1519/492, 4920/1519, -3344161/747348)
48sage: a,b,c=congruent_triangle(5*P)
49sage: a
50_21 = -394091011800472369443/63458283116489076790
51sage: b
52_22 = -634582831164890767900/394091011800472369443
53sage: c
54_23 = -160443526614433014168714029147613242401001/25008339000498013289668468371318876527970
55sage: E
56_24 = Elliptic Curve defined by y^2  = x^3 - 25*x over Rational Field
57sage: factor(conductor(E))
58_25 = 2^5 * 5^2
59sage: cong_nu
60cong_number_curve  cong_number_sets
61sage: cong_number_sets?
62Type:           function
63Base Class:     <type 'function'>
64String Form:    <function cong_number_sets at 0xaacedd84>
65Namespace:      Interactive
66File:           /home/was/edu/fall05/168/notes/2005-10-17/cong.sage.py
67Definition:     cong_number_sets(n)
68Docstring:
69    Given a positive integer n, returns the two sets appearing in the
70    conjectural criterion for when a number is congruent.
71
72sage: cong_number_sets(6)
73_27 = ([], [])
74sage: cong_number_sets(5)
75_28 = ([], [])
76sage: S1, S2 = cong_number_sets(7)
77sage: S1
78_30 = []
79sage: S2
80_31 = []
81sage: S1, S2 = cong_number_sets(100)
82sage: S1
83_33 = [(0, 1, 0)]
84sage: S2
85_34 = []
86sage: S1, S2 = cong_number_sets(1)
87sage: S1
88_36 = [(0, 1, 0)]
89sage: S2
90_37 = []
91sage: S1, S2 = cong_number_sets(103)
92sage: S1
93_39 = []
94sage: S2
95_40 = []
96sage: E = con
97conductor                   cong_number_sets
98cong.sage                   congruent_curve_gens
99cong.sage.aux               congruent_triangle
100cong.sage.dvi               conj_congruent_number_list
101cong.sage.log               continue
102cong.sage.pdf               continued_fraction
103cong.sage.ps                convergent
104cong.sage.py                convergents
105cong.sage.tex               conway_polynomial
106cong_number_curve
107sage: E = cong_nu
108cong_number_curve  cong_number_sets
109sage: E = cong_number_curve(103)
110sage: E
111_42 = Elliptic Curve defined by y^2  = x^3 - 10609*x over Rational Field
112sage: G = E.gens()
113sage: G
114_44 = [(-777848715219380607/8780605285453456, 406939902409963977921570495/822785599723202981879104)]
115sage: P = G[0]
116sage: P
117_46 = (-777848715219380607/8780605285453456, 406939902409963977921570495/822785599723202981879104)
118sage: congru
119congruent_curve_gens  congruent_triangle
120sage: congruent_triangle(P)
121_47 =
122(45463628564396045/8143126555471908,
123 16286253110943816/441394452081515,
124 134130664938047228374702001079697/3594330884182957394223708580620)
125sage: congruent_triangle(2*P)
126_48 =
127(17185634719391093727634843473743585469340755051722944799891950559/964219983005638593080701999601243903601008247213261446739344280,
128 198629316499161550174624611917856244141807698925931858028304921680/17185634719391093727634843473743585469340755051722944799891950559,
129 -352008659948220111075691074151631552714265844032730144831000926906517193179085542508749236648581510050936484589962917565740092481/16570732417072392965427706076087110093738148708575603533511700446069960034049208047977307349747461389650861475528706577839452520)
130sage: S1, S2 = cong_number_sets(541)
131sage: S1
132_50 = []
133sage: S2
134_51 = []
135sage: v = [n for n in range(1,200) if is_conj_congruent_number(n)]
136sage: print v
137[5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, 124, 125, 126, 127, 133, 134, 135, 136, 137, 138, 141, 142, 143, 145, 148, 149, 150, 151, 152, 154, 156, 157, 158, 159, 161, 164, 165, 166, 167, 173, 174, 175, 180, 181, 182, 183, 184, 188, 189, 190, 191, 194, 197, 198, 199]
138sage: set([a%8 for a in v])
139_54 = set([1, 0, 2, 5, 4, 7, 6])
140sage: v = [n for n in range(200,250) if is_conj_congruent_number(n)]
141sage: set([a%8 for a in v])
142_56 = set([0, 3, 2, 5, 4, 7, 6])
143sage: is_conj_congruent_number(219)
144_57 = True
145sage: is_conj_congruent_number(2005)
146_58 = True
147sage: E = cong_number_curve(2005)
148sage: E
149_60 = Elliptic Curve defined by y^2  = x^3 - 4020025*x over Rational Field
150sage: G = E.gens(verbose=True)
151
1523 points of order 2:
153[0:0:1], [2005:0:1], [-2005:0:1]
154
155****************************
156* Using 2-isogeny number 1 *
157****************************
158
159Using 2-isogenous curve [0,0,0,16080100,0]
160-------------------------------------------------------
161First step, determining 1st descent Selmer groups
162-------------------------------------------------------
163After first local descent, rank bound = 3
164rk(S^{phi}(E'))=        3
165rk(S^{phi'}(E))=        2
166
167-------------------------------------------------------
168Second step, determining 2nd descent Selmer groups
169-------------------------------------------------------
170After second local descent, rank bound = 3
171rk(phi'(S^{2}(E)))=     3
172rk(phi(S^{2}(E')))=     2
173rk(S^{2}(E))=   5
174rk(S^{2}(E'))=  4
175
176****************************
177* Using 2-isogeny number 2 *
178****************************
179
180Using 2-isogenous curve [0,-12030,0,4020025,0]
181-------------------------------------------------------
182First step, determining 1st descent Selmer groups
183-------------------------------------------------------
184After first local descent, rank bound = 3
185rk(S^{phi}(E'))=        4
186rk(S^{phi'}(E))=        1
187
188-------------------------------------------------------
189Second step, determining 2nd descent Selmer groups
190-------------------------------------------------------
191After second local descent, rank bound = 3
192rk(phi'(S^{2}(E)))=     4
193rk(phi(S^{2}(E')))=     1
194rk(S^{2}(E))=   5
195rk(S^{2}(E'))=  4
196
197****************************
198* Using 2-isogeny number 3 *
199****************************
200
201Using 2-isogenous curve [0,12030,0,4020025,0]
202-------------------------------------------------------
203First step, determining 1st descent Selmer groups
204-------------------------------------------------------
205After first local descent, rank bound = 3
206rk(S^{phi}(E'))=        3
207rk(S^{phi'}(E))=        2
208
209-------------------------------------------------------
210Second step, determining 2nd descent Selmer groups
211-------------------------------------------------------
212After second local descent, rank bound = 2
213rk(phi'(S^{2}(E)))=     3
214rk(phi(S^{2}(E')))=     1
215rk(S^{2}(E))=   5
216rk(S^{2}(E'))=  3
217
218After second local descent, combined upper bound on rank = 2
219Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
220-------------------------------------------------------
2211. E(Q)/phi(E'(Q))
222-------------------------------------------------------
223(c,d)  =(-6015,8040050)
224(c',d')=(12030,4020025)
225First stage (no second descent yet)...
226(5,0,-6015,0,1608010):  no rational point found (hlim=8)
227(10,0,-6015,0,804005):  no rational point found (hlim=8)
228(401,0,-6015,0,20050):  no rational point found (hlim=8)
229(802,0,-6015,0,10025):  no rational point found (hlim=8)
230After first global descent, this component of the rank
231        has lower bound 0
232        and upper bound 1
233        (difference =   1)
234Second descent will attempt to reduce this
235Second stage (using second descent)...
236d1=5:
237 (x:y:z) = (12883:337361925:158)
238        Curve E         Point [131117634310:21731168398875:3944312], height = 18.893783589929163211674233870788384718695982504872
239Second descent successfully found rational point for d1=5
240
241After second global descent, this component of the rank = 3
242-------------------------------------------------------
2432. E'(Q)/phi'(E(Q))
244-------------------------------------------------------
245First stage (no second descent yet)...
246(401,0,12030,0,10025):  no rational point found (hlim=8)
247After first global descent, this component of the rank
248        has lower bound 0
249        and upper bound 1
250        (difference =   1)
251Second descent will attempt to reduce this
252Second stage (using second descent)...
253d1=401:
254Second descent inconclusive for d1=401: ELS descendents exist but no rational point found
255
256After second global descent, this component of the rank
257        has lower bound 0
258        and upper bound 1
259        (difference =   1)
260
261-------------------------------------------------------
262Summary of results:
263-------------------------------------------------------
264        1 <= rank(E) <= 2
265        #E(Q)/2E(Q) >= 8
266
267Information on III(E/Q):
268        #III(E/Q)[phi']    <= 4
269        #III(E/Q)[2]       is between 2 and 4
270
271Information on III(E'/Q):
272        #phi'(III(E/Q)[2]) = 1
273        #III(E'/Q)[phi]    = 1
274        #III(E'/Q)[2]      <= 2
275
276-------------------------------------------------------
277
278List of points on E = [0,0,0,-4020025,0]:
279
280I.  Points on E mod phi(E')
281Point [123209288750:21731168398875:3944312], height = 18.893783589929163211674233870788384718695982504871
282
283II.  Points on phi(E') mod 2E
284--none (modulo torsion).
285
286-------------------------------------------------------
287Computing full set of 2 coset representatives for
2882E(Q) in E(Q) (modulo torsion), and sorting into height order....done.
289
290
291Regulator (before saturation) = 18.893783589929163211674233870788384718695982504871
292Searching for points (bound = 10)...done
293Regulator (after searching) = 18.893783589929163211674233870788384718695982504871
294Saturating (bound = -1)...finished saturation (index was 1)
295Regulator (after saturation) = 18.893783589929163211674233870788384718695982504871
296Unable to compute the rank, hence generators, with certainty (lower bound=1).  This could be because Sha(E/Q)[2] is nontrivial.
297sage: G
298_62 = [(779805625/24964, 21731168398875/3944312)]
299sage: P = G[0]
300sage: a,b,c=congruent_triangle(P)
301sage: a
302_65 = -155639523/882430
303sage:  b
304_66 = -3538544300/155639523
305sage: c
306_67 = 24424083902863721/137340984280890
307sage: factor(2005)
308_68 = 5 * 401
309sage: factor(2006)
310_69 = 2 * 17 * 59
311sage: E = cong_number_curve(2006)
312sage: E
313_71 = Elliptic Curve defined by y^2  = x^3 - 4024036*x over Rational Field
314sage: is_conj_congruent_number(2006)
315_72 = True
316sage: E = cong_number_curve(2006)
317sage: G = E.gens(verbose=True)
318
3193 points of order 2:
320[0:0:1], [2006:0:1], [-2006:0:1]
321
322****************************
323* Using 2-isogeny number 1 *
324****************************
325
326Using 2-isogenous curve [0,0,0,16096144,0]
327-------------------------------------------------------
328First step, determining 1st descent Selmer groups
329-------------------------------------------------------
330After first local descent, rank bound = 3
331rk(S^{phi}(E'))=        4
332rk(S^{phi'}(E))=        1
333
334-------------------------------------------------------
335Second step, determining 2nd descent Selmer groups
336-------------------------------------------------------
337After second local descent, rank bound = 3
338rk(phi'(S^{2}(E)))=     4
339rk(phi(S^{2}(E')))=     1
340rk(S^{2}(E))=   5
341rk(S^{2}(E'))=  4
342
343****************************
344* Using 2-isogeny number 2 *
345****************************
346
347Using 2-isogenous curve [0,-12036,0,4024036,0]
348-------------------------------------------------------
349First step, determining 1st descent Selmer groups
350-------------------------------------------------------
351After first local descent, rank bound = 3
352rk(S^{phi}(E'))=        4
353rk(S^{phi'}(E))=        1
354
355-------------------------------------------------------
356Second step, determining 2nd descent Selmer groups
357-------------------------------------------------------
358After second local descent, rank bound = 3
359rk(phi'(S^{2}(E)))=     4
360rk(phi(S^{2}(E')))=     1
361rk(S^{2}(E))=   5
362rk(S^{2}(E'))=  4
363
364****************************
365* Using 2-isogeny number 3 *
366****************************
367
368Using 2-isogenous curve [0,12036,0,4024036,0]
369-------------------------------------------------------
370First step, determining 1st descent Selmer groups
371-------------------------------------------------------
372After first local descent, rank bound = 3
373rk(S^{phi}(E'))=        3
374rk(S^{phi'}(E))=        2
375
376-------------------------------------------------------
377Second step, determining 2nd descent Selmer groups
378-------------------------------------------------------
379After second local descent, rank bound = 2
380rk(phi'(S^{2}(E)))=     3
381rk(phi(S^{2}(E')))=     1
382rk(S^{2}(E))=   5
383rk(S^{2}(E'))=  3
384
385After second local descent, combined upper bound on rank = 2
386Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
387-------------------------------------------------------
3881. E(Q)/phi(E'(Q))
389-------------------------------------------------------
390(c,d)  =(-6018,8048072)
391(c',d')=(12036,4024036)
392First stage (no second descent yet)...
393(17,0,-6018,0,473416):  (x:y:z) = (10:204:1)
394        Curve E         Point [1700:34680:1], height = 3.7970846911387271852102431813863172782263003517029
395(59,0,-6018,0,136408):  (x:y:z) = (3:295:1)
396        Curve E         Point [531:52215:1], height = 3.7970846911387271852102431813863172782263003517028
397After first global descent, this component of the rank = 3
398-------------------------------------------------------
3992. E'(Q)/phi'(E(Q))
400-------------------------------------------------------
401First stage (no second descent yet)...
402(17,0,12036,0,236708):  no rational point found (hlim=8)
403After first global descent, this component of the rank
404        has lower bound 0
405        and upper bound 1
406        (difference =   1)
407Second descent will attempt to reduce this
408Second stage (using second descent)...
409d1=17:
410Second descent inconclusive for d1=17: ELS descendents exist but no rational point found
411
412After second global descent, this component of the rank
413        has lower bound 0
414        and upper bound 1
415        (difference =   1)
416
417-------------------------------------------------------
418Summary of results:
419-------------------------------------------------------
420        1 <= rank(E) <= 2
421        #E(Q)/2E(Q) >= 8
422
423Information on III(E/Q):
424        #III(E/Q)[phi']    <= 4
425        #III(E/Q)[2]       is between 2 and 4
426
427Information on III(E'/Q):
428        #phi'(III(E/Q)[2]) = 1
429        #III(E'/Q)[phi]    = 1
430        #III(E'/Q)[2]      <= 2
431
432-------------------------------------------------------
433
434List of points on E = [0,0,0,-4024036,0]:
435
436I.  Points on E mod phi(E')
437Point [-306:34680:1], height = 3.7970846911387271852102431813863172782263003517032
438Point [-1475:52215:1], height = 3.7970846911387271852102431813863172782263003517033
439
440II.  Points on phi(E') mod 2E
441--none (modulo torsion).
442
443-------------------------------------------------------
444Computing full set of 2 coset representatives for
4452E(Q) in E(Q) (modulo torsion), and sorting into height order....done.
446
447
448Regulator (before saturation) = 3.7970846911387271852102431813863172782263003517032
449Searching for points (bound = 10)...done
450Regulator (after searching) = 3.7970846911387271852102431813863172782263003517032
451Saturating (bound = -1)...finished saturation (index was 1)
452Regulator (after saturation) = 3.7970846911387271852102431813863172782263003517032
453Unable to compute the rank, hence generators, with certainty (lower bound=1).  This could be because Sha(E/Q)[2] is nontrivial.
454sage: G
455_75 = [(-306, 34680)]
456sage: P = G[0]
457sage: a,b,c=congruent_triangle(P)
458sage: a,b,c
459_78 = (340/3, 177/5, 1781/15)
460sage: is_conj_congruent_number(2007)
461_79 = True
462sage: is_conj_congruent_number(2008)
463_80 = True
464sage: is_conj_congruent_number(2009)
465_81 = True
466sage: is_conj_congruent_number(2010)
467_82 = False
468sage: time v = [n for n in range(1, 300) if is_conj_congruent_number(n)]
469CPU times: user 6.91 s, sys: 0.32 s, total: 7.24 s
470Wall time: 7.43
471sage: len(v)
472_84 = 163
473sage:
474