was@form:~$ cd 168/notes/2005-10-17
was@form:~/168/notes/2005-10-17$ sage
------------------------------------------------------------------------
  SAGE Version 0.7.13, Export Date: 2005-10-17-0407
  Distributed under the terms of the GNU General Public License (GPL)
  IPython shell -- for help type <object>?, <object>??, %magic, or help
------------------------------------------------------------------------

sage: (3/2)^2 + (20/3)^2
 _1 = 1681/36
sage: (41/6)^2
 _2 = 1681/36
sage: attach "cong.sage"
sage: E = congru
congruent_curve_gens  congruent_triangle
sage: E = cong_num
cong_number_curve  cong_number_sets
sage: E = cong_number_curve(5)
sage: E
 _5 = Elliptic Curve defined by y^2  = x^3 - 25*x over Rational Field
sage: G = E.gens()
sage: G
 _7 = [(-4, 6)]
sage: P = G[0]
sage: P
 _9 = (-4, 6)
sage: congrue
congruent_curve_gens  congruent_triangle
sage: congruent_triangle(P)
_10 = (3/2, 20/3, 41/6)
sage: 2*P
_11 = (1681/144, -62279/1728)
sage: congruent_triangle(2*P)
_12 = (1519/492, 4920/1519, -3344161/747348)
sage: a,b,c = _12
sage: a
_14 = 1519/492
sage: b
_15 = 4920/1519
sage: c
_16 = -3344161/747348
sage: a^2 + b^2 == c^2
_17 = True
sage: a*b/2
_18 = 5
sage: a,b,c
_19 = (1519/492, 4920/1519, -3344161/747348)
sage: a,b,c=congruent_triangle(5*P)
sage: a
_21 = -394091011800472369443/63458283116489076790
sage: b
_22 = -634582831164890767900/394091011800472369443
sage: c
_23 = -160443526614433014168714029147613242401001/25008339000498013289668468371318876527970
sage: E
_24 = Elliptic Curve defined by y^2  = x^3 - 25*x over Rational Field
sage: factor(conductor(E))
_25 = 2^5 * 5^2
sage: cong_nu
cong_number_curve  cong_number_sets
sage: cong_number_sets?
Type:           function
Base Class:     <type 'function'>
String Form:    <function cong_number_sets at 0xaacedd84>
Namespace:      Interactive
File:           /home/was/edu/fall05/168/notes/2005-10-17/cong.sage.py
Definition:     cong_number_sets(n)
Docstring:
    Given a positive integer n, returns the two sets appearing in the
    conjectural criterion for when a number is congruent.

sage: cong_number_sets(6)
_27 = ([], [])
sage: cong_number_sets(5)
_28 = ([], [])
sage: S1, S2 = cong_number_sets(7)
sage: S1
_30 = []
sage: S2
_31 = []
sage: S1, S2 = cong_number_sets(100)
sage: S1
_33 = [(0, 1, 0)]
sage: S2
_34 = []
sage: S1, S2 = cong_number_sets(1)
sage: S1
_36 = [(0, 1, 0)]
sage: S2
_37 = []
sage: S1, S2 = cong_number_sets(103)
sage: S1
_39 = []
sage: S2
_40 = []
sage: E = con
conductor                   cong_number_sets
cong.sage                   congruent_curve_gens
cong.sage.aux               congruent_triangle
cong.sage.dvi               conj_congruent_number_list
cong.sage.log               continue
cong.sage.pdf               continued_fraction
cong.sage.ps                convergent
cong.sage.py                convergents
cong.sage.tex               conway_polynomial
cong_number_curve
sage: E = cong_nu
cong_number_curve  cong_number_sets
sage: E = cong_number_curve(103)
sage: E
_42 = Elliptic Curve defined by y^2  = x^3 - 10609*x over Rational Field
sage: G = E.gens()
sage: G
_44 = [(-777848715219380607/8780605285453456, 406939902409963977921570495/822785599723202981879104)]
sage: P = G[0]
sage: P
_46 = (-777848715219380607/8780605285453456, 406939902409963977921570495/822785599723202981879104)
sage: congru
congruent_curve_gens  congruent_triangle
sage: congruent_triangle(P)
_47 =
(45463628564396045/8143126555471908,
 16286253110943816/441394452081515,
 134130664938047228374702001079697/3594330884182957394223708580620)
sage: congruent_triangle(2*P)
_48 =
(17185634719391093727634843473743585469340755051722944799891950559/964219983005638593080701999601243903601008247213261446739344280,
 198629316499161550174624611917856244141807698925931858028304921680/17185634719391093727634843473743585469340755051722944799891950559,
 -352008659948220111075691074151631552714265844032730144831000926906517193179085542508749236648581510050936484589962917565740092481/16570732417072392965427706076087110093738148708575603533511700446069960034049208047977307349747461389650861475528706577839452520)
sage: S1, S2 = cong_number_sets(541)
sage: S1
_50 = []
sage: S2
_51 = []
sage: v = [n for n in range(1,200) if is_conj_congruent_number(n)]
sage: print v
[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]
sage: set([a%8 for a in v])
_54 = set([1, 0, 2, 5, 4, 7, 6])
sage: v = [n for n in range(200,250) if is_conj_congruent_number(n)]
sage: set([a%8 for a in v])
_56 = set([0, 3, 2, 5, 4, 7, 6])
sage: is_conj_congruent_number(219)
_57 = True
sage: is_conj_congruent_number(2005)
_58 = True
sage: E = cong_number_curve(2005)
sage: E
_60 = Elliptic Curve defined by y^2  = x^3 - 4020025*x over Rational Field
sage: G = E.gens(verbose=True)

3 points of order 2:
[0:0:1], [2005:0:1], [-2005:0:1]

****************************
* Using 2-isogeny number 1 *
****************************

Using 2-isogenous curve [0,0,0,16080100,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 3
rk(S^{phi}(E'))=        3
rk(S^{phi'}(E))=        2

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 3
rk(phi'(S^{2}(E)))=     3
rk(phi(S^{2}(E')))=     2
rk(S^{2}(E))=   5
rk(S^{2}(E'))=  4

****************************
* Using 2-isogeny number 2 *
****************************

Using 2-isogenous curve [0,-12030,0,4020025,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 3
rk(S^{phi}(E'))=        4
rk(S^{phi'}(E))=        1

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 3
rk(phi'(S^{2}(E)))=     4
rk(phi(S^{2}(E')))=     1
rk(S^{2}(E))=   5
rk(S^{2}(E'))=  4

****************************
* Using 2-isogeny number 3 *
****************************

Using 2-isogenous curve [0,12030,0,4020025,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 3
rk(S^{phi}(E'))=        3
rk(S^{phi'}(E))=        2

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 2
rk(phi'(S^{2}(E)))=     3
rk(phi(S^{2}(E')))=     1
rk(S^{2}(E))=   5
rk(S^{2}(E'))=  3

After second local descent, combined upper bound on rank = 2
Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d)  =(-6015,8040050)
(c',d')=(12030,4020025)
First stage (no second descent yet)...
(5,0,-6015,0,1608010):  no rational point found (hlim=8)
(10,0,-6015,0,804005):  no rational point found (hlim=8)
(401,0,-6015,0,20050):  no rational point found (hlim=8)
(802,0,-6015,0,10025):  no rational point found (hlim=8)
After first global descent, this component of the rank
        has lower bound 0
        and upper bound 1
        (difference =   1)
Second descent will attempt to reduce this
Second stage (using second descent)...
d1=5:
 (x:y:z) = (12883:337361925:158)
        Curve E         Point [131117634310:21731168398875:3944312], height = 18.893783589929163211674233870788384718695982504872
Second descent successfully found rational point for d1=5

After second global descent, this component of the rank = 3
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
First stage (no second descent yet)...
(401,0,12030,0,10025):  no rational point found (hlim=8)
After first global descent, this component of the rank
        has lower bound 0
        and upper bound 1
        (difference =   1)
Second descent will attempt to reduce this
Second stage (using second descent)...
d1=401:
Second descent inconclusive for d1=401: ELS descendents exist but no rational point found

After second global descent, this component of the rank
        has lower bound 0
        and upper bound 1
        (difference =   1)

-------------------------------------------------------
Summary of results:
-------------------------------------------------------
        1 <= rank(E) <= 2
        #E(Q)/2E(Q) >= 8

Information on III(E/Q):
        #III(E/Q)[phi']    <= 4
        #III(E/Q)[2]       is between 2 and 4

Information on III(E'/Q):
        #phi'(III(E/Q)[2]) = 1
        #III(E'/Q)[phi]    = 1
        #III(E'/Q)[2]      <= 2

-------------------------------------------------------

List of points on E = [0,0,0,-4020025,0]:

I.  Points on E mod phi(E')
Point [123209288750:21731168398875:3944312], height = 18.893783589929163211674233870788384718695982504871

II.  Points on phi(E') mod 2E
--none (modulo torsion).

-------------------------------------------------------
Computing full set of 2 coset representatives for
2E(Q) in E(Q) (modulo torsion), and sorting into height order....done.


Regulator (before saturation) = 18.893783589929163211674233870788384718695982504871
Searching for points (bound = 10)...done
Regulator (after searching) = 18.893783589929163211674233870788384718695982504871
Saturating (bound = -1)...finished saturation (index was 1)
Regulator (after saturation) = 18.893783589929163211674233870788384718695982504871
Unable to compute the rank, hence generators, with certainty (lower bound=1).  This could be because Sha(E/Q)[2] is nontrivial.
sage: G
_62 = [(779805625/24964, 21731168398875/3944312)]
sage: P = G[0]
sage: a,b,c=congruent_triangle(P)
sage: a
_65 = -155639523/882430
sage:  b
_66 = -3538544300/155639523
sage: c
_67 = 24424083902863721/137340984280890
sage: factor(2005)
_68 = 5 * 401
sage: factor(2006)
_69 = 2 * 17 * 59
sage: E = cong_number_curve(2006)
sage: E
_71 = Elliptic Curve defined by y^2  = x^3 - 4024036*x over Rational Field
sage: is_conj_congruent_number(2006)
_72 = True
sage: E = cong_number_curve(2006)
sage: G = E.gens(verbose=True)

3 points of order 2:
[0:0:1], [2006:0:1], [-2006:0:1]

****************************
* Using 2-isogeny number 1 *
****************************

Using 2-isogenous curve [0,0,0,16096144,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 3
rk(S^{phi}(E'))=        4
rk(S^{phi'}(E))=        1

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 3
rk(phi'(S^{2}(E)))=     4
rk(phi(S^{2}(E')))=     1
rk(S^{2}(E))=   5
rk(S^{2}(E'))=  4

****************************
* Using 2-isogeny number 2 *
****************************

Using 2-isogenous curve [0,-12036,0,4024036,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 3
rk(S^{phi}(E'))=        4
rk(S^{phi'}(E))=        1

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 3
rk(phi'(S^{2}(E)))=     4
rk(phi(S^{2}(E')))=     1
rk(S^{2}(E))=   5
rk(S^{2}(E'))=  4

****************************
* Using 2-isogeny number 3 *
****************************

Using 2-isogenous curve [0,12036,0,4024036,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 3
rk(S^{phi}(E'))=        3
rk(S^{phi'}(E))=        2

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 2
rk(phi'(S^{2}(E)))=     3
rk(phi(S^{2}(E')))=     1
rk(S^{2}(E))=   5
rk(S^{2}(E'))=  3

After second local descent, combined upper bound on rank = 2
Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d)  =(-6018,8048072)
(c',d')=(12036,4024036)
First stage (no second descent yet)...
(17,0,-6018,0,473416):  (x:y:z) = (10:204:1)
        Curve E         Point [1700:34680:1], height = 3.7970846911387271852102431813863172782263003517029
(59,0,-6018,0,136408):  (x:y:z) = (3:295:1)
        Curve E         Point [531:52215:1], height = 3.7970846911387271852102431813863172782263003517028
After first global descent, this component of the rank = 3
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
First stage (no second descent yet)...
(17,0,12036,0,236708):  no rational point found (hlim=8)
After first global descent, this component of the rank
        has lower bound 0
        and upper bound 1
        (difference =   1)
Second descent will attempt to reduce this
Second stage (using second descent)...
d1=17:
Second descent inconclusive for d1=17: ELS descendents exist but no rational point found

After second global descent, this component of the rank
        has lower bound 0
        and upper bound 1
        (difference =   1)

-------------------------------------------------------
Summary of results:
-------------------------------------------------------
        1 <= rank(E) <= 2
        #E(Q)/2E(Q) >= 8

Information on III(E/Q):
        #III(E/Q)[phi']    <= 4
        #III(E/Q)[2]       is between 2 and 4

Information on III(E'/Q):
        #phi'(III(E/Q)[2]) = 1
        #III(E'/Q)[phi]    = 1
        #III(E'/Q)[2]      <= 2

-------------------------------------------------------

List of points on E = [0,0,0,-4024036,0]:

I.  Points on E mod phi(E')
Point [-306:34680:1], height = 3.7970846911387271852102431813863172782263003517032
Point [-1475:52215:1], height = 3.7970846911387271852102431813863172782263003517033

II.  Points on phi(E') mod 2E
--none (modulo torsion).

-------------------------------------------------------
Computing full set of 2 coset representatives for
2E(Q) in E(Q) (modulo torsion), and sorting into height order....done.


Regulator (before saturation) = 3.7970846911387271852102431813863172782263003517032
Searching for points (bound = 10)...done
Regulator (after searching) = 3.7970846911387271852102431813863172782263003517032
Saturating (bound = -1)...finished saturation (index was 1)
Regulator (after saturation) = 3.7970846911387271852102431813863172782263003517032
Unable to compute the rank, hence generators, with certainty (lower bound=1).  This could be because Sha(E/Q)[2] is nontrivial.
sage: G
_75 = [(-306, 34680)]
sage: P = G[0]
sage: a,b,c=congruent_triangle(P)
sage: a,b,c
_78 = (340/3, 177/5, 1781/15)
sage: is_conj_congruent_number(2007)
_79 = True
sage: is_conj_congruent_number(2008)
_80 = True
sage: is_conj_congruent_number(2009)
_81 = True
sage: is_conj_congruent_number(2010)
_82 = False
sage: time v = [n for n in range(1, 300) if is_conj_congruent_number(n)]
CPU times: user 6.91 s, sys: 0.32 s, total: 7.24 s
Wall time: 7.43
sage: len(v)
_84 = 163
sage: