sage_notebook/ 0000777 0001750 0001750 00000000000 10474617506 012061 5 ustar was was sage_notebook/tmp/ 0000777 0001750 0001750 00000000000 10474617506 012661 5 ustar was was sage_notebook/nb-backup.sobj 0000755 0001750 0001750 00001347246 10474617463 014624 0 ustar was was csage.server.notebook.notebook
Notebook
q)q}q(U_Notebook__worksheetsq}q(Ubsdq(csage.server.notebook.worksheet
Worksheet
qoq}q (U_Worksheet__filenameq
UbsdqU_Worksheet__cellsq]q
((csage.server.notebook.cell
Cell
qoq}q(U _Cell__inqT@ %hide%html
The Birch and Swinnerton-Dyer Conjecture for 37A
This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
for the elliptic curve
y^2 + y = x^3 - x,
affectionately referred to by mathematicians as 37a.
qU_Cell__introspect_htmlqU!qU_Cell__worksheetqhU_Cell__completionsqU_Cell__introspectqU_Cell__out_htmlqU U _Cell__idqMU_before_preparseqT os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/3")
%hide%html
The Birch and Swinnerton-Dyer Conjecture for 37A
This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
for the elliptic curve
y^2 + y = x^3 - x,
affectionately referred to by mathematicians as 37a.
qU
_Cell__dirqU$sage_notebook/worksheets/bsd/cells/3qU
_Cell__outqTf
The Birch and Swinnerton-Dyer Conjecture for 37A
This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
for the elliptic curve
y^2 + y = x^3 - x,
affectionately referred to by mathematicians as 37a.
qUhas_new_outputq U_Cell__is_htmlq!U_Cell__sageq"csage.interfaces.sage0
reduce_load_Sage
q#)Rq$U_Cell__typeq%Uwrapq&U_Cell__timeq'U_Cell__interruptedq(ub(hoq)}q*(U _Cell__inq+T %hide%html
An elliptic curve
E
over
\mathbf{Q}
is a smooth projective curve determined by an equation
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6,
with
a_1,a_2,a_3,a_4,a_6 \in \mathbf{Q}.
EXAMPLE:
y^2 + y = x^3 - x
q,U_Cell__introspect_htmlq-U!q.U_Cell__worksheetq/hU_Cell__completionsq0U_Cell__introspectq1U_Cell__out_htmlq2U U _Cell__idq3MU_before_preparseq4T: os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/24")
%hide%html
An elliptic curve
E
over
\mathbf{Q}
is a smooth projective curve determined by an equation
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6,
with
a_1,a_2,a_3,a_4,a_6 \in \mathbf{Q}.
EXAMPLE:
y^2 + y = x^3 - x
q5U
_Cell__dirq6U%sage_notebook/worksheets/bsd/cells/24q7U
_Cell__outq8T
An elliptic curve
E
over
\mathbf{Q}
is a smooth projective curve determined by an equation
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6,
with
a_1,a_2,a_3,a_4,a_6 \in \mathbf{Q}.
EXAMPLE:
y^2 + y = x^3 - x
q9Uhas_new_outputq:U_Cell__is_htmlq;U_Cell__sageqU_Cell__interruptedq?ub(hoq@}qA(hUZE = EllipticCurve([0,0,1,-1,0])
G = plot(E, rgbcolor=(1,0,1), thickness=3)
show(G,dpi=160)qBhU!qChhhhhU<qDhM hUos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/0")
E = EllipticCurve([0,0,1,-1,0])
G = plot(E, rgbcolor=(1,0,1), thickness=3)
show(G,dpi=160)qEhU$sage_notebook/worksheets/bsd/cells/0qFhU h h!h"h$h%h&h'h(ub(hoqG}qH(hUoE = EllipticCurve([0,0,1,-1,0])
P = E([0,0])
Q = P
for n in range(30):
print '%-10s%-60s'%(n, Q)
Q += PqIhU!qJhhhhhU hMhUos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/8")
E = EllipticCurve([0,0,1,-1,0])
P = E([0,0])
Q = P
for n in range(30):
print '%-10s%-60s'%(n, Q)
Q += PqKhU$sage_notebook/worksheets/bsd/cells/8qLhTn 0 (0 : 0 : 1)
1 (1 : 0 : 1)
2 (-1 : -1 : 1)
3 (2 : -3 : 1)
4 (1/4 : -5/8 : 1)
5 (6 : 14 : 1)
6 (-5/9 : 8/27 : 1)
7 (21/25 : -69/125 : 1)
8 (-20/49 : -435/343 : 1)
9 (161/16 : -2065/64 : 1)
10 (116/529 : -3612/12167 : 1)
11 (1357/841 : 28888/24389 : 1)
12 (-3741/3481 : -43355/205379 : 1)
13 (18526/16641 : -2616119/2146689 : 1)
14 (8385/98596 : -28076979/30959144 : 1)
15 (480106/4225 : 332513754/274625 : 1)
16 (-239785/2337841 : 331948240/3574558889 : 1)
17 (12551561/13608721 : -8280062505/50202571769 : 1)
18 (-59997896/67387681 : -641260644409/553185473329 : 1)
19 (683916417/264517696 : -18784454671297/4302115807744 : 1)
20 (1849037896/6941055969 : -318128427505160/578280195945297 : 1)
21 (51678803961/12925188721 : 10663732503571536/1469451780501769 : 1)
22 (-270896443865/384768368209 : 66316334575107447/238670664494938073 : 1)
23 (4881674119706/5677664356225 : -8938035295591025771/13528653463047586625 : 1)
24 (-16683000076735/61935294530404 : -588310630753491921045/487424450554237378792 : 1)
25 (997454379905326/49020596163841 : -31636113722016288336230/343216282443844010111 : 1)
26 (2786836257692691/16063784753682169 : -435912379274109872312968/2035972062206737347698803 : 1)
27 (213822353304561757/158432514799144041 : 41974401721854929811774227/63061816101171948456692661 : 1)
28 (-3148929681285740316/2846153597907293521 : -2181616293371330311419201915/4801616835579099275862827431 : 1)
29 (79799551268268089761/62586636021357187216 : -754388827236735824355996347601/495133617181351428873673516736 : 1)qMh h!h"h$h%h&h'h(ub(hoqN}qO(hUE = EllipticCurve([0,0,1,-1,0])
G = plot(E, thickness=.6, rgbcolor=(1,0,1))
Q = P
n = 100
for i in range(n):
Q = Q + P
if abs(Q[0]) < 3 and abs(Q[1]) < 5:
G += point(Q,rgbcolor=(1,0,0),pointsize=10+float(i)*100/n)
show(G,dpi=150)qPhU!qQhhhhhU=qRhMhTK os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/11")
E = EllipticCurve([0,0,1,-1,0])
G = plot(E, thickness=.6, rgbcolor=(1,0,1))
Q = P
n = 100
for i in range(n):
Q = Q + P
if abs(Q[0]) < 3 and abs(Q[1]) < 5:
G += point(Q,rgbcolor=(1,0,0),pointsize=10+float(i)*100/n)
show(G,dpi=150)qShU%sage_notebook/worksheets/bsd/cells/11qThU h h!h"h$h%h&h'h(ub(hoqU}qV(hTA t = Tachyon(xres=1000, yres=800, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
t.light((10,3,2), 1, (1,1,1))
t.light((10,-3,2), 1, (1,1,1))
t.texture('black', color=(0,0,0))
t.texture('red', color=(1,0,0))
t.texture('grey', color=(.9,.9,.9))
t.plane((0,0,0),(0,0,1),'grey')
t.cylinder((0,0,0),(1,0,0),.01,'black')
t.cylinder((0,0,0),(0,1,0),.01,'black')
E = EllipticCurve('37a')
P = E([0,0])
Q = P
n = 100
for i in range(n):
Q = Q + P
c = i/n + .1
t.texture('r%s'%i,color=(float(i/n),0,0))
t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i)
t.save()qWhU!qXhhhhhU;qYhMhT os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/6")
t = Tachyon(xres=1000, yres=800, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
t.light((10,3,2), 1, (1,1,1))
t.light((10,-3,2), 1, (1,1,1))
t.texture('black', color=(0,0,0))
t.texture('red', color=(1,0,0))
t.texture('grey', color=(.9,.9,.9))
t.plane((0,0,0),(0,0,1),'grey')
t.cylinder((0,0,0),(1,0,0),.01,'black')
t.cylinder((0,0,0),(0,1,0),.01,'black')
E = EllipticCurve('37a')
P = E([0,0])
Q = P
n = 100
for i in range(n):
Q = Q + P
c = i/n + .1
t.texture('r%s'%i,color=(float(i/n),0,0))
t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i)
t.save()qZhU$sage_notebook/worksheets/bsd/cells/6q[hU h h!h"h$h%h&h'h(ub(hoq\}q](hT\ %hide%html
A Sharp Contrast:
-
The elliptic curve
y^2 + y = x^3 - x
has infinitely many rational points.
-
The elliptic curve
y^2 + y = x^3 - x^2
has only finitely many rational points!
Question 1: Is there an a priori way to tell which type
of elliptic curve we are dealing with?
Question 2: How often does each possibility occur? (Conjecture: 50% each -- See Bektemirov, Mazur, Stein, Watkins)
q^hU!q_hhhhhU hMhT os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/7")
%hide%html
A Sharp Contrast:
-
The elliptic curve
y^2 + y = x^3 - x
has infinitely many rational points.
-
The elliptic curve
y^2 + y = x^3 - x^2
has only finitely many rational points!
Question 1: Is there an a priori way to tell which type
of elliptic curve we are dealing with?
Question 2: How often does each possibility occur? (Conjecture: 50% each -- See Bektemirov, Mazur, Stein, Watkins)
q`hU$sage_notebook/worksheets/bsd/cells/7qahT
A Sharp Contrast:
-
The elliptic curve
y^2 + y = x^3 - x
has infinitely many rational points.
-
The elliptic curve
y^2 + y = x^3 - x^2
has only finitely many rational points!
Question 1: Is there an a priori way to tell which type
of elliptic curve we are dealing with?
Question 2: How often does each possibility occur? (Conjecture: 50% each -- See Bektemirov, Mazur, Stein, Watkins)
qbh h!h"h$h%h&h'h(ub(hoqc}qd(hT %hide%html
The L-function
of an elliptic curve is a
function on the complex numbers
defined by counting points modulo primes:
\displaystyle
L(E,s) = \prod_{p} \left(\frac{1}{1-a_p p^{-s} + p^{1-2s}}\right).
Formally:
\displaystyle
L(E,1) = \prod_{p} \frac{p}{\#E(\mathbf{F}_p)}.
qehU!qfhhhhhU hM
hT os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/13")
%hide%html
The L-function
of an elliptic curve is a
function on the complex numbers
defined by counting points modulo primes:
\displaystyle
L(E,s) = \prod_{p} \left(\frac{1}{1-a_p p^{-s} + p^{1-2s}}\right).
Formally:
\displaystyle
L(E,1) = \prod_{p} \frac{p}{\#E(\mathbf{F}_p)}.
qghU%sage_notebook/worksheets/bsd/cells/13qhhT
The L-function
of an elliptic curve is a
function on the complex numbers
defined by counting points modulo primes:
\displaystyle
L(E,s) = \prod_{p} \left(\frac{1}{1-a_p p^{-s} + p^{1-2s}}\right).
Formally:
\displaystyle
L(E,1) = \prod_{p} \frac{p}{\#E(\mathbf{F}_p)}.
qih h!h"h$h%h&h'h(ub(hoqj}qk(hU# Some Pictures of Counting Points
E = EllipticCurve([0,0,1,-1,0])
G = [plot(E.change_ring(GF(p)), pointsize=30, rgbcolor=(1,0,0))\
for p in primes(42) if p!=37]
show(graphics_array(G,4,3),fontsize=4)qlhU!qmhhhhhU>qnhMhT$ os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/14")
# Some Pictures of Counting Points
E = EllipticCurve([0,0,1,-1,0])
G = [plot(E.change_ring(GF(p)), pointsize=30, rgbcolor=(1,0,0)) for p in primes(42) if p!=37]
show(graphics_array(G,4,3),fontsize=4)qohU%sage_notebook/worksheets/bsd/cells/14qphU h h!h"h$h%h&h'h(ub(hoqq}qr(hU# Tally up the number of points (including point at infinity)
E = EllipticCurve([0,0,1,-1,0])
print '.'*40
print '%10s%-2s%10s%13s'%('','p','N_p', 'p+1-N_p')
for p in primes(1000):
print '%10s%-10s%-10s%-10s'%('',p,E.Np(p),E.ap(p))qshU!qthhhhhU hMhTB os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/16")
# Tally up the number of points (including point at infinity)
E = EllipticCurve([0,0,1,-1,0])
print '.'*40
print '%10s%-2s%10s%13s'%('','p','N_p', 'p+1-N_p')
for p in primes(1000):
print '%10s%-10s%-10s%-10s'%('',p,E.Np(p),E.ap(p))quhU%sage_notebook/worksheets/bsd/cells/16qvhT ........................................
p N_p p+1-N_p
2 5 -2
3 7 -3
5 8 -2
7 9 -1
11 17 -5
13 16 -2
17 18 0
19 20 0
23 22 2
29 24 6
31 36 -4
37 39 -1
41 51 -9
43 42 2
47 57 -9
53 53 1
59 52 8
61 70 -8
67 60 8
71 63 9
73 75 -1
79 76 4
83 99 -15
89 86 4
97 94 4
101 99 3
103 86 18
107 120 -12
109 126 -16
113 132 -18
127 127 1
131 144 -12
137 144 -6
139 136 4
149 155 -5
151 136 16
157 135 23
163 182 -18
167 180 -12
173 165 9
179 162 18
181 177 5
191 196 -4
193 220 -26
197 195 3
199 198 2
211 225 -13
223 241 -17
227 244 -16
229 223 7
233 228 6
239 246 -6
241 228 14
251 254 -2
257 258 0
263 245 19
269 276 -6
271 303 -31
277 266 12
281 270 12
283 280 4
293 296 -2
307 325 -17
311 312 0
313 292 22
317 296 22
331 334 -2
337 363 -25
347 358 -10
349 344 6
353 346 8
359 375 -15
367 360 8
373 393 -19
379 365 15
383 364 20
389 386 4
397 403 -5
401 384 18
409 390 20
419 413 7
421 446 -24
431 462 -30
433 425 9
439 412 28
443 443 1
449 414 36
457 440 18
461 432 30
463 486 -22
467 470 -2
479 466 14
487 512 -24
491 520 -28
499 488 12
503 488 16
509 541 -31
521 555 -33
523 546 -22
541 522 20
547 540 8
557 576 -18
563 594 -30
569 594 -24
571 565 7
577 578 0
587 620 -32
593 599 -5
599 599 1
601 624 -22
607 640 -32
613 599 15
617 601 17
619 621 -1
631 660 -28
641 643 -1
643 630 14
647 656 -8
653 678 -24
659 675 -15
661 690 -28
673 647 27
677 689 -11
683 666 18
691 712 -20
701 714 -12
709 670 40
719 681 39
727 712 16
733 727 7
739 749 -9
743 723 21
751 727 25
757 808 -50
761 797 -35
769 744 26
773 783 -9
787 793 -5
797 746 52
809 808 2
811 765 47
821 869 -47
823 840 -16
827 806 22
829 834 -4
839 796 44
853 828 26
857 906 -48
859 880 -20
863 888 -24
877 828 50
881 896 -14
883 836 48
887 863 25
907 856 52
911 886 26
919 978 -58
929 912 18
937 901 37
941 952 -10
947 936 12
953 893 61
967 982 -14
971 980 -8
977 950 28
983 975 9
991 1010 -18
997 1040 -42qwh h!h"h$h%h&h'h(ub(hoqx}qy(h+Tr %hide
print """
ASIDE -- big recent theorem of Taylor, Harris,
Clozel, Shepherd-Barron
THE SATO-TATE CONJECTURE is a theorem
(in a wide range of cases).
"""
#%auto
def dist(v, b, left=float(0), right=float(pi)):
"""
We divide the interval between left (default: 0) and
right (default: pi) up into b bins.
For each number in v (which must left and right),
we find which bin it lies in and add this to a counter.
This function then returns the bins and the number of
elements of v that lie in each one.
ALGORITHM: To find the index of the bin that a given
number x lies in, we multiply x by b/length and take the
floor.
"""
length = right - left
normalize = float(b/length)
vals = {}
d = dict([(i,0) for i in range(b)])
for x in v:
n = int(normalize*(float(x)-left))
d[n] += 1
return d, len(v)
def graph(d, b, num=5000, left=float(0), right=float(pi)):
s = Graphics()
left = float(left); right = float(right)
length = right - left
w = length/b
k = 0
for i, n in d.iteritems():
k += n
# ith bin has n objects in it.
s += polygon([(w*i+left,0), (w*(i+1)+left,0), \
(w*(i+1)+left, n/(num*w)), \
(w*i+left, n/(num*w))],\
rgbcolor=(0,0,0.5))
return s
def sato_tate(E, N):
return [E.ap(p)/(2*sqrt(p)) for p in prime_range(N+1) \
if N%p != 0]
def graph_ellcurve(E, b=10, num=5000):
v = sato_tate(E, num)
d, total_number_of_points = dist(v,b,-1,1)
return graph(d, b, total_number_of_points,-1,1)
C = plot( lambda x: (2/pi) * sqrt(1-x^2), -1, 1, \
plot_points=200, \
rgbcolor=(1,0,0), thickness=4,alpha=0.7)
G = graph_ellcurve(E,50,num=50000)
show(C + G, ymin=0, ymax=.75, figsize=[8,4])qzh-U!q{h/hh0h1h2U>q|h3Mh4T os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/23")
%hide
print """
ASIDE -- big recent theorem of Taylor, Harris,
Clozel, Shepherd-Barron
THE SATO-TATE CONJECTURE is a theorem
(in a wide range of cases).
"""
#%auto
def dist(v, b, left=float(0), right=float(pi)):
"""
We divide the interval between left (default: 0) and
right (default: pi) up into b bins.
For each number in v (which must left and right),
we find which bin it lies in and add this to a counter.
This function then returns the bins and the number of
elements of v that lie in each one.
ALGORITHM: To find the index of the bin that a given
number x lies in, we multiply x by b/length and take the
floor.
"""
length = right - left
normalize = float(b/length)
vals = {}
d = dict([(i,0) for i in range(b)])
for x in v:
n = int(normalize*(float(x)-left))
d[n] += 1
return d, len(v)
def graph(d, b, num=5000, left=float(0), right=float(pi)):
s = Graphics()
left = float(left); right = float(right)
length = right - left
w = length/b
k = 0
for i, n in d.iteritems():
k += n
# ith bin has n objects in it.
s += polygon([(w*i+left,0), (w*(i+1)+left,0), (w*(i+1)+left, n/(num*w)), (w*i+left, n/(num*w))],\
rgbcolor=(0,0,0.5))
return s
def sato_tate(E, N):
return [E.ap(p)/(2*sqrt(p)) for p in prime_range(N+1) if N%p != 0]
def graph_ellcurve(E, b=10, num=5000):
v = sato_tate(E, num)
d, total_number_of_points = dist(v,b,-1,1)
return graph(d, b, total_number_of_points,-1,1)
C = plot( lambda x: (2/pi) * sqrt(1-x^2), -1, 1, plot_points=200, rgbcolor=(1,0,0), thickness=4,alpha=0.7)
G = graph_ellcurve(E,50,num=50000)
show(C + G, ymin=0, ymax=.75, figsize=[8,4])q}h6U%sage_notebook/worksheets/bsd/cells/23q~h8U
ASIDE -- big recent theorem of Taylor, Harris,
Clozel, Shepherd-Barron
THE SATO-TATE CONJECTURE is a theorem
(in a wide range of cases).
qh:h;hh?ub(hoq}q(hTy %hide%html
Recall that
the L-series of E
is
\displaystyle
L(E,s) = \prod_{p} \left(\frac{1}{1-a_p p^{-s} + p^{1-2s}}\right).
where
a_p = p+1 - \#E(\mathbf{Z}/p\mathbf{Z})
qhU!qhhhhhU hMU_word_being_completedqUE.Lseries_doqhT os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/17")
%hide%html
Recall that
the L-series of E
is
\displaystyle
L(E,s) = \prod_{p} \left(\frac{1}{1-a_p p^{-s} + p^{1-2s}}\right).
where
a_p = p+1 - \#E(\mathbf{Z}/p\mathbf{Z})
qhU%sage_notebook/worksheets/bsd/cells/17qhT
Recall that
the L-series of E
is
\displaystyle
L(E,s) = \prod_{p} \left(\frac{1}{1-a_p p^{-s} + p^{1-2s}}\right).
where
a_p = p+1 - \#E(\mathbf{Z}/p\mathbf{Z})
qh h!h"h$h%h&h'h(ub(hoq}q(hU# Compute the L-series of E
E = EllipticCurve([0,0,1,-1,0])
L = E.Lseries_dokchitser(10) # Tim Dokchitser
plot(L, -2,3, rgbcolor=(0,0,1), plot_points=90, \
plot_division=0, thickness=2).show()qhU!qhhhhhU=qhMhT! os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/18")
# Compute the L-series of E
E = EllipticCurve([0,0,1,-1,0])
L = E.Lseries_dokchitser(10) # Tim Dokchitser
plot(L, -2,3, rgbcolor=(0,0,1), plot_points=90, plot_division=0, thickness=2).show()qhU%sage_notebook/worksheets/bsd/cells/18qhU h h!h"h$h%h&h'h(ub(hoq}q(hT{ t = Tachyon(xres=800, yres=600, camera_center=(1.2,.4,.4), look_at=(1,0,0), raydepth=2)
t.light((10,3,2), 1, (1,1,1))
t.light((10,-3,2), 1, (1,1,1))
t.texture('black', color=(0,0,0))
t.texture('red', color=(1,0,0))
t.texture('white', color=(1,1,1))
t.plane((0,0,-10),(0,0,1),'white')
t.cylinder((0,0,0),(1,0,0),.001,'black')
t.cylinder((0,0,0),(0,1,0),.001,'black')
n=1000
for i in range(n):
x = random()/2+.8; y = random()/2 - .25
try:
z = L(x+I*y)
m = abs(z)
r = arg(z)+pi
except:
continue
t.texture('r%s'%i,color=(r/7,r,0))
t.sphere((x,-y,m), .009, 'r%s'%i)
t.show()qhU!qhhhhhU>qhMhUrandoqhT os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/19")
t = Tachyon(xres=800, yres=600, camera_center=(1.2,.4,.4), look_at=(1,0,0), raydepth=2)
t.light((10,3,2), 1, (1,1,1))
t.light((10,-3,2), 1, (1,1,1))
t.texture('black', color=(0,0,0))
t.texture('red', color=(1,0,0))
t.texture('white', color=(1,1,1))
t.plane((0,0,-10),(0,0,1),'white')
t.cylinder((0,0,0),(1,0,0),.001,'black')
t.cylinder((0,0,0),(0,1,0),.001,'black')
n=1000
for i in range(n):
x = random()/2+.8; y = random()/2 - .25
try:
z = L(x+I*y)
m = abs(z)
r = arg(z)+pi
except:
continue
t.texture('r%s'%i,color=(r/7,r,0))
t.sphere((x,-y,m), .009, 'r%s'%i)
t.show()qhU%sage_notebook/worksheets/bsd/cells/19qhU h h!h"h$h%h&h'h(ub(hoq}q(h+T| t = Tachyon(xres=800, yres=600, camera_center=(1.2,.4,.4), look_at=(1,0,0), raydepth=2)
t.light((10,3,2), 1, (1,1,1))
t.light((10,-3,2), 1, (1,1,1))
t.texture('black', color=(0,0,0))
t.texture('red', color=(1,0,0))
t.texture('white', color=(1,1,1))
t.plane((0,0,-10),(0,0,1),'white')
t.cylinder((0,0,0),(1,0,0),.001,'black')
t.cylinder((0,0,0),(0,1,0),.001,'black')
n=10000
for i in range(n):
x = random()/2+.8; y = random()/2 - .25
try:
z = L(x+I*y)
m = abs(z)
r = arg(z)+pi
except:
continue
t.texture('r%s'%i,color=(r/7,r,0))
t.sphere((x,-y,m), .005, 'r%s'%i)
t.show()qh-U!qh/hh0h1h2U>qh3Mh4T os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/26")
t = Tachyon(xres=800, yres=600, camera_center=(1.2,.4,.4), look_at=(1,0,0), raydepth=2)
t.light((10,3,2), 1, (1,1,1))
t.light((10,-3,2), 1, (1,1,1))
t.texture('black', color=(0,0,0))
t.texture('red', color=(1,0,0))
t.texture('white', color=(1,1,1))
t.plane((0,0,-10),(0,0,1),'white')
t.cylinder((0,0,0),(1,0,0),.001,'black')
t.cylinder((0,0,0),(0,1,0),.001,'black')
n=10000
for i in range(n):
x = random()/2+.8; y = random()/2 - .25
try:
z = L(x+I*y)
m = abs(z)
r = arg(z)+pi
except:
continue
t.texture('r%s'%i,color=(r/7,r,0))
t.sphere((x,-y,m), .005, 'r%s'%i)
t.show()qh6U%sage_notebook/worksheets/bsd/cells/26qh8U h:h;hh?ub(hoq}q(U _Cell__inqT print """
Birch's Parallel Lines?
"""
E = [EllipticCurve('11a'), EllipticCurve('37a'),
EllipticCurve('389a'), EllipticCurve('5077a')]
def f(E, B=1000, **args):
v = []; pr = 1
for p in prime_range(2,B):
pr *= float(p/E.Np(p))
if p >= 5:
v.append((p, pr))
return line(v, **args) + point(v,**args)
G = sum([f(E[i],rgbcolor=(i/4.0,0,1-i/4.0)) for i in range(4)])
show(G,ymin=0,ymax=.25,dpi=200)
qU_Cell__introspect_htmlqU!qU_Cell__worksheetqhU_Cell__completionsqU_Cell__introspectqU_Cell__out_htmlqU>qU _Cell__idqMU_before_preparseqT7 os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/27")
print """
Birch's Parallel Lines?
"""
E = [EllipticCurve('11a'), EllipticCurve('37a'),
EllipticCurve('389a'), EllipticCurve('5077a')]
def f(E, B=1000, **args):
v = []; pr = 1
for p in prime_range(2,B):
pr *= float(p/E.Np(p))
if p >= 5:
v.append((p, pr))
return line(v, **args) + point(v,**args)
G = sum([f(E[i],rgbcolor=(i/4.0,0,1-i/4.0)) for i in range(4)])
show(G,ymin=0,ymax=.25,dpi=200)qU
_Cell__dirqU%sage_notebook/worksheets/bsd/cells/27qU
_Cell__outqUC
Birch's Parallel Lines?
qUhas_new_outputqU_Cell__is_htmlqU_Cell__sageqh$U_Cell__typeqh&U_Cell__timeqU_Cell__interruptedqub(hoq}q(hT %hide%html
Conjecture (Birch and Swinnerton-Dyer):
\text{ord}_{s=1} L(E,s) = \text{rank } E
- This is a theorem
when the order of vanishing is 0 or 1.
- It is a million dollar Clay Problem.
-
It is an open problem
to prove that the order of vanishing
can ever be 4 or larger.
-
There is also a beautiful conjecture
(that is very useful computationally!)
about the first nonzero coefficient
of the Taylor expansion of L(E,s)
about s=1.
(back to slides)qhU!qhhhhhU hMhT[ os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/bsd/cells/21")
%hide%html
Conjecture (Birch and Swinnerton-Dyer):
\text{ord}_{s=1} L(E,s) = \text{rank } E
- This is a theorem
when the order of vanishing is 0 or 1.
- It is a million dollar Clay Problem.
-
It is an open problem
to prove that the order of vanishing
can ever be 4 or larger.
-
There is also a beautiful conjecture
(that is very useful computationally!)
about the first nonzero coefficient
of the Taylor expansion of L(E,s)
about s=1.
(back to slides)qhU%sage_notebook/worksheets/bsd/cells/21qhT=
Conjecture (Birch and Swinnerton-Dyer):
\text{ord}_{s=1} L(E,s) = \text{rank } E
- This is a theorem
when the order of vanishing is 0 or 1.
- It is a million dollar Clay Problem.
-
It is an open problem
to prove that the order of vanishing
can ever be 4 or larger.
-
There is also a beautiful conjecture
(that is very useful computationally!)
about the first nonzero coefficient
of the Taylor expansion of L(E,s)
about s=1.
(back to slides)qh h!h"h$h%h&h'h(ub(hoq}q(hU hhhhU hMhU%sage_notebook/worksheets/bsd/cells/22qhU h h%h&h(ubeU_Worksheet__synchroqMmU_Worksheet__comp_is_runningqU_Worksheet__dirqUsage_notebook/worksheets/bsdqU_Worksheet__attachedq}qU_Worksheet__queueq]qU_Worksheet__next_idqMU_Worksheet__nameqUbsdqU_Worksheet__notebookqhU_Worksheet__idqKU_Worksheet__next_block_idqK*U_Worksheet__systemqNubU _scratch_q(hoq}q(U_Worksheet__filenameqU _scratch_qU_Worksheet__cellsq]q((hoq}q(h+Ulatex(EllipticCurve('11a3'))qh-U!qh/hh0h1h2U h3K-h4Uxos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/45")
latex(EllipticCurve('11a3'))qh6U+sage_notebook/worksheets/_scratch_/cells/45qh8Uy^2 + y = x^3 - x^2qh:h;hh?ub(hoq}q(h+Uqh/hh0h1h2U h3K(h4Uos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/40")
E = EllipticCurve([0,0,1,-1,0])
L = E.Lseries_dokchitser(10)qh6U+sage_notebook/worksheets/_scratch_/cells/40qh8U h:h;hh?ub(hoq}q(h+Uz1 = L(1-0.1*I); z1qh-U!qh/hh0h1h2U h3K)h4Uoos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/41")
z1 = L(1-0.1*I); z1qh6U+sage_notebook/worksheets/_scratch_/cells/41qh8U-0.0018635 - 0.030731*Iqh:h;hh?ub(hoq}q(h+Uz2 = L(1+0.1*I); z2qh-U!qh/hh0h1h2U h3K*h4Uoos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/42")
z2 = L(1+0.1*I); z2qh6U+sage_notebook/worksheets/_scratch_/cells/42qh8U-0.0018635 + 0.030731*Iqh:h;hh?ub(hoq}q(h+Uabs(z1), arg(z1)qh-U!qh/hh0h1h2U h3K+h4Ulos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/43")
abs(z1), arg(z1)qh6U+sage_notebook/worksheets/_scratch_/cells/43qh8U(0.030792, -1.6309)qh:h;hh?ub(hoq}q(h+Uabs(z2), arg(z2)qh-U!qh/hh0h1h2U h3K,h4Ulos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/44")
abs(z2), arg(z2)qh6U+sage_notebook/worksheets/_scratch_/cells/44qh8U(0.030792, 1.6309)r h:h;hh?ub(hor }r (U _Cell__inr Uarg(2-.001*I) + pir U_Cell__introspect_htmlr U!r U_Cell__worksheetr hU_Cell__completionsr U_Cell__introspectr U_Cell__out_htmlr
U U _Cell__idr K U_before_preparser Upos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/0")
arg(2-.001*I) + pir
U
_Cell__dirr U*sage_notebook/worksheets/_scratch_/cells/0r U
_Cell__outr U3.1410926536314596r Uhas_new_outputr U_Cell__is_htmlr U_Cell__sager h$U_Cell__typer hU_Cell__timer U_Cell__interruptedr ub(hor }r (h+Uarg(-(2-.001*I))r h-U!r h/hh0h1h2U h3K'h4Ulos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/39")
arg(-(2-.001*I))r h6U+sage_notebook/worksheets/_scratch_/cells/39r h8U3.1410926536314601r h:h;hh?ub(hor }r (h+U
I.argumentr! h-T File: /Volumes/HOME/s/local/lib/python2.4/site-packages/sage/rings/complex_number.py
Source Code (starting at line 497):
def argument(self):
r"""
The argument (angle) of the complex number, normalized
so that $-\pi < \theta \leq \pi$.
EXAMPLES:
sage: i = CC.0
sage: (i^2).argument()
3.1415926535897931
sage: (1+i).argument()
0.78539816339744828
sage: i.argument()
1.5707963267948966
sage: (-i).argument()
-1.5707963267948966
sage: (RR('-0.001') - i).argument()
-1.5717963264615635
"""
return self.parent()(self._pari_().arg())
r" h/hh0h1]r# (UI.argument??r$ U eh2U h3K&U_word_being_completedr% UI.argumer& h4Uhos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/38")
I.argument??r' h6U+sage_notebook/worksheets/_scratch_/cells/38r( h8U h:h;h=hh>h?ub(hor) }r* (h+U%slide
An {\dred elliptic curve} $E$ over $\mathbf{Q}$
is a smooth projective curve defined by an equation
$$
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6,
$$
with $a_1,a_2,a_3,a_4,a_6 \in \mathbf{Q}$.r+ h-U!r, h/hh0h1h2UFr- h3K%j% Uhtml.evr. h4T. os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/37")
%slide
An {\dred elliptic curve} $E$ over $\mathbf{Q}$
is a smooth projective curve defined by an equation
$$
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6,
$$
with $a_1,a_2,a_3,a_4,a_6 \in \mathbf{Q}$.r/ h6U+sage_notebook/worksheets/_scratch_/cells/37r0 h8U h:h;hh?ub(hor1 }r2 (U _Cell__inr3 U%python
print(2^3)
print(2/5)r4 U_Cell__introspect_htmlr5 U!r6 U_Cell__worksheetr7 hU_Cell__completionsr8 U_Cell__introspectr9 U_Cell__out_htmlr: U U _Cell__idr; KU_before_preparser< Uzos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/21")
%python
print(2^3)
print(2/5)r= U
_Cell__dirr> U+sage_notebook/worksheets/_scratch_/cells/21r? U
_Cell__outr@ U1
0rA Uhas_new_outputrB U_Cell__is_htmlrC U_Cell__sagerD h$U_Cell__typerE UwraprF U_Cell__timerG U_Cell__interruptedrH ub(horI }rJ (j Ufactor(2007)rK j U!rL j hj j j
U j Kj Ugos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/1")
factor(2007)rM j U*sage_notebook/worksheets/_scratch_/cells/1rN j U 3^2 * 223rO j j j h$j hj j ub(horP }rQ (j U5factor(2^997-1) # you can hit escape and it works!rR j U!rS j hj j j
U j Kj Uos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/5")
factor(2^997-1) # you can hit escape and it works!rT j U*sage_notebook/worksheets/_scratch_/cells/5rU j U j j j Nj UwraprV j j ub(horW }rX (U _Cell__inrY U/M = MatrixSpace(QQ,20).random_element()
show(M)rZ U_Cell__introspect_htmlr[ U!r\ U_Cell__worksheetr] hU_Cell__completionsr^ U_Cell__introspectr_ U_Cell__out_htmlr` U U _Cell__idra KU_before_preparserb Uos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/12")
M = MatrixSpace(QQ,20).random_element()
show(M)rc U
_Cell__dirrd U+sage_notebook/worksheets/_scratch_/cells/12re U
_Cell__outrf T~ \left(\begin{array}{rrrrrrrrrrrrrrrrrrrr}
-2&1&-1&1&1&2&1&1&1&1&-1&2&-1&-1&1&2&-1&-1&1&2\\
1&-2&1&-1&-2&-2&-1&-2&-1&-2&-1&-2&1&-2&-1&-1&-2&-1&1&1\\
-1&2&-2&1&1&1&-2&2&2&-2&1&-1&1&2&1&1&1&2&1&-1\\
-1&-1&2&-2&-1&1&-2&-2&-1&-1&2&-1&1&-2&-1&1&2&2&2&2\\
1&1&-1&2&-2&1&1&1&2&-1&-2&-1&1&1&-2&1&-1&-2&-2&-1\\
-1&-1&-2&-1&-2&-1&1&1&-1&2&1&1&-1&-2&2&1&-2&-2&-2&-2\\
2&2&1&-2&-2&1&2&1&-1&2&2&1&-1&-1&2&1&2&-1&-1&2\\
2&2&1&-1&2&1&-2&1&-2&-1&-1&-1&-1&-2&-1&1&2&-1&1&-2\\
-1&-2&-1&2&1&-2&2&-2&1&-2&1&1&-2&1&2&2&-2&1&1&-2\\
-2&2&-2&2&1&-1&1&2&1&-2&2&-2&1&-2&2&1&2&-1&2&-1\\
-2&2&2&-2&1&-2&1&1&-2&1&1&2&1&-1&-1&-2&-2&2&-2&-1\\
2&1&-2&2&1&-1&-2&1&1&-1&1&2&1&-1&-2&2&-1&-2&-1&-1\\
-2&2&-2&1&2&2&-2&-2&1&1&-1&1&1&-2&1&-1&-2&1&-1&1\\
-1&1&2&-1&1&1&1&-2&-2&-1&-1&-1&-2&-2&-1&1&-1&1&-2&2\\
-2&-1&1&-2&1&-1&-2&-2&1&-2&2&2&-2&2&-1&1&2&-2&-1&2\\
1&1&-2&2&2&-2&-2&-1&-2&1&2&-1&1&-1&1&2&-2&2&-2&-2\\
1&2&-2&1&-2&2&-1&2&-2&2&-1&-2&-2&2&1&2&-1&-1&1&1\\
1&2&2&2&-2&-2&1&-1&-1&-1&1&-1&-2&-1&2&-2&1&2&-2&2\\
1&2&-2&-1&-1&-2&-2&2&2&2&-1&-2&1&2&1&1&-2&2&2&1\\
-2&2&-1&-2&-1&2&2&1&1&-1&-1&1&2&-1&2&1&-2&1&1&2
\end{array}\right)
rg Uhas_new_outputrh U_Cell__is_htmlri U_Cell__sagerj h$U_Cell__typerk Uwraprl U_Cell__timerm U_Cell__interruptedrn ub(horo }rp (j3 U
M.parent()rq j5 U!rr j7 hj8 j9 j: U j; Kj< Ufos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/24")
M.parent()rs j> U+sage_notebook/worksheets/_scratch_/cells/24rt j@ U=Full MatrixSpace of 3 by 3 dense matrices over Rational Fieldru jB jC jD h$jE jV jG jH ub(horv }rw (jY Uview(M)rx j[ U!ry j] hj^ j_ j` U ja K
jb Ucos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/13")
view(M)rz jd U+sage_notebook/worksheets/_scratch_/cells/13r{ jf U\left(\begin{array}{rrr}
1&2&3\\
\frac{1}{3}&17&-\frac{2}{3}\\
1&5&-5
\end{array}\right)r| jh ji jj h$jk jV jm jn ub(hor} }r~ (j3 Uview(M.echelon_form())r j5 U!r j7 hj8 j9 j: U j; KU_word_being_completedr UM.echr j< Uros.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/20")
view(M.echelon_form())r j> U+sage_notebook/worksheets/_scratch_/cells/20r j@ Ul\left(\begin{array}{rrr}
1&0&0\\
0&1&0\\
0&0&1
\end{array}\right)r jB jC jD h$jE jV jG jH ub(hor }r (jY UG = AlternatingGroup(5)r j[ U!r j] hj^ j_ j` U ja KU_word_being_completedr USymmetrir jb Upos.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/14")
G = AlternatingGroup(5)r jd U+sage_notebook/worksheets/_scratch_/cells/14r jf U jh ji jj h$jk Uwrapr jm jn ub(hor }r (jY U%G.conjugacy_classes_representatives()r j[ U!r j] hj^ j_ j` U ja Kj UG.conjugr jb U~os.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/15")
G.conjugacy_classes_representatives()r jd U+sage_notebook/worksheets/_scratch_/cells/15r jf U3[(), (1,2)(3,4), (1,2,3), (1,2,3,4,5), (1,2,3,5,4)]r jh ji jj h$jk j jm jn ub(hor }r (jY UG%time
f = x^389 + 17/3*x + 2
g = x^397 - 18*x + 15
h = f*g^10 + f^10*gr j[ U!r j] hj^ j_ j` U ja Kjb Uos.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/16")
__SAGE_t__=cputime()
__SAGE_w__=walltime()
f = x^389 + 17/3*x + 2
g = x^397 - 18*x + 15
h = f*g^10 + f^10*gr jd U+sage_notebook/worksheets/_scratch_/cells/16r jf U$CPU time: 0.02 s, Wall time: 0.02 sr jh ji jj h$jk j jm jn ub(hor }r (U _Cell__inr Uh[:50]r U_Cell__introspect_htmlr U!r U_Cell__worksheetr hU_Cell__completionsr U_Cell__introspectr U_Cell__out_htmlr U U _Cell__idr KU_before_preparser U_os.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/28")
h[:50]r U
_Cell__dirr U+sage_notebook/worksheets/_scratch_/cells/28r U
_Cell__outr Tv [1153300796610, -10571923411357, 35521669089060, -82207173890380/3, -1578378351911680/9, 6297739398214880/9, -107392893078774688/81, 377409713492725760/243, -287199547816076060/243, 3757883697935134460/6561, -3178137624289098691/19683, 132742369030852798/6561, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]r Uhas_new_outputr U_Cell__is_htmlr U_Cell__sager h$U_Cell__typer j U_Cell__timer U_Cell__interruptedr ub(hor }r (jY Uview(f)r j[ U!r j] hj^ j_ j` U ja Kjb U`os.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/17")
view(f)r jd U+sage_notebook/worksheets/_scratch_/cells/17r jf UBx^{389} + \frac{17}{3}x + 2r jh ji jj h$jk j jm jn ub(hor }r (U _Cell__inr Ug%hide%html
You can embed HTML
It can even include math: \prod(1-q^n)r U_Cell__introspect_htmlr U!r U_Cell__worksheetr hU_Cell__completionsr U_Cell__introspectr U_Cell__out_htmlr U U _Cell__idr K U_before_preparser Uos.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/9")
%hide%html
You can embed HTML
It can even include math: \prod(1-q^n)r U
_Cell__dirr U*sage_notebook/worksheets/_scratch_/cells/9r U
_Cell__outr UjYou can embed HTML
It can even include math: \prod(1-q^n)r Uhas_new_outputr U_Cell__is_htmlr U_Cell__sager h$U_Cell__typer j U_Cell__timer U_Cell__interruptedr ub(hor }r (j Uu%latex
Cells can be written in latex, which can refer to SAGE objects.
For example, consider $E$ given by $\sage{E}$.r j U!r j hj j j UEr j Kj Uos.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/11")
%latex
Cells can be written in latex, which can refer to SAGE objects.
For example, consider $E$ given by $\sage{E}$.r j U+sage_notebook/worksheets/_scratch_/cells/11r j U j j j h$j j j j ub(hor }r (j Utime n=factorial(10^6)r j U!r j hj j j
U j Kj Uos.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/3")
__SAGE_t__=cputime()
__SAGE_w__=walltime()
n=factorial(10^6)r j U*sage_notebook/worksheets/_scratch_/cells/3r j U$CPU time: 3.31 s, Wall time: 3.34 sr j j j h$j j j j ub(hor }r (j U0f = maxima('x*sin(x)*cos(x)^2')
print f, type(f)r j U!r j hj j j
U j Kj Uos.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/4")
f = maxima('x*sin(x)*cos(x)^2')
print f, type(f)r j U*sage_notebook/worksheets/_scratch_/cells/4r j U@x*cos(x)^2*sin(x) r j j j h$j j j j ub(hor }r (j3 U
f.integrate()r j5 U!r j7 hj8 j9 j: U j; Kj Uf.r j< Ufos.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/22")
f.integrate()r j> U+sage_notebook/worksheets/_scratch_/cells/22r j@ U4(sin(3*x) - 3*x*cos(3*x) + 9*sin(x) - 9*x*cos(x))/36r jB jC jD h$jE j jG jH ub(hor }r (j3 Uf.name()r j5 U!r j7 hj8 j9 j: U j; Kj< Uaos.chdir("/home/was/talks/2006-08-17-scipy/sage_notebook/worksheets/_scratch_/cells/23")
f.name()r j> U+sage_notebook/worksheets/_scratch_/cells/23r j@ U'sage0'r jB jC jD h$jE Uhiddenr jG jH ub(hor }r (hU;E = EllipticCurve('37a')
show(plot(E, rgbcolor=(1,0,0)))
Er hU!r hhhhhUDr hKhUos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/31")
E = EllipticCurve('37a')
show(plot(E, rgbcolor=(1,0,0)))
Er hU+sage_notebook/worksheets/_scratch_/cells/31r hU?Elliptic Curve defined by y^2 + y = x^3 - x over Rational Fieldr h h!h"h$h%jV h'h(ub(hor }r (j T t = Tachyon(xres=1000, yres=800, camera_center=(0,5,7), look_at=(-.5,1,0), raydepth=4)
t.light((10,3,2), 0.2, (1,1,1))
t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
t.texture('white', color=(1,1,1))
t.texture('black', color=(1,1,1))
t.texture('grey', color=(.9,.9,.9))
#t.plane((0,-100000,0),(0,1,0),'white')
#t.plane((-100000,0,0),(1,0,0),'white')
t.plane((-100000,0,0),(0,0,1),'grey')
t.cylinder((0,0,0),(1,0,0),.1,'black')
t.cylinder((0,0,0),(0,1,0),.1,'black')
E = EllipticCurve('37a')
show(plot(E))
P = E([0,0])
Q = P
n = 200
for i in range(n):
Q = Q + P
c = i/(10*n) + .1
t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
t.save() r j5 U!r j hj j9 j
U
r j KhUt.cylr j< T os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/6")
t = Tachyon(xres=1000, yres=800, camera_center=(0,5,7), look_at=(-.5,1,0), raydepth=4)
t.light((10,3,2), 0.2, (1,1,1))
t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
t.texture('white', color=(1,1,1))
t.texture('black', color=(1,1,1))
t.texture('grey', color=(.9,.9,.9))
#t.plane((0,-100000,0),(0,1,0),'white')
#t.plane((-100000,0,0),(1,0,0),'white')
t.plane((-100000,0,0),(0,0,1),'grey')
t.cylinder((0,0,0),(1,0,0),.1,'black')
t.cylinder((0,0,0),(0,1,0),.1,'black')
E = EllipticCurve('37a')
show(plot(E))
P = E([0,0])
Q = P
n = 200
for i in range(n):
Q = Q + P
c = i/(10*n) + .1
t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
t.save()r j U*sage_notebook/worksheets/_scratch_/cells/6r j U j jC jD h$j jV jG j ub(hor }r (hT E = EllipticCurve('37a')
L = E.Lseries_dokchitser(5)
t = Tachyon(xres=800, yres=600, camera_center=(1.5,.4,.7), look_at=(1,0,0), raydepth=4)
t.light((10,3,2), 1, (1,1,1))
t.light((10,-3,2), 1, (1,1,1))
t.texture('black', color=(0,0,0))
t.texture('red', color=(1,0,0))
t.texture('grey', color=(.9,.9,.9))
t.plane((0,0,0),(0,0,1),'grey')
t.cylinder((0,0,0),(1,0,0),.01,'black')
t.cylinder((0,0,0),(0,1,0),.01,'black')
def f(x,y):
return abs(L(x+I*y))r
hU!r hhhhhU hKhT os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/30")
E = EllipticCurve('37a')
L = E.Lseries_dokchitser(5)
t = Tachyon(xres=800, yres=600, camera_center=(1.5,.4,.7), look_at=(1,0,0), raydepth=4)
t.light((10,3,2), 1, (1,1,1))
t.light((10,-3,2), 1, (1,1,1))
t.texture('black', color=(0,0,0))
t.texture('red', color=(1,0,0))
t.texture('grey', color=(.9,.9,.9))
t.plane((0,0,0),(0,0,1),'grey')
t.cylinder((0,0,0),(1,0,0),.01,'black')
t.cylinder((0,0,0),(0,1,0),.01,'black')
def f(x,y):
return abs(L(x+I*y))r hU+sage_notebook/worksheets/_scratch_/cells/30r
hU h h!h"h$h%jV h'h(ub(hor }r (hU hhhhU hK!hU+sage_notebook/worksheets/_scratch_/cells/33r hU h h%jV h(ub(hor }r (hUE = EllipticCurve('11a')r hU!r hhhhhU hK hUtos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/32")
E = EllipticCurve('11a')r hU+sage_notebook/worksheets/_scratch_/cells/32r hU h h!h"h$h%jV h'h(ub(hor }r (hUtime v = E.Lseries_zeros(100)r hU!r hhhhhU hK"hUE.Lseries_zr hUos.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/34")
__SAGE_t__=cputime()
__SAGE_w__=walltime()
v = E.Lseries_zeros(100)r hU+sage_notebook/worksheets/_scratch_/cells/34r hU$CPU time: 0.01 s, Wall time: 8.14 sr h h!h"h$h%jV h'h(ub(hor }r (hUvhU!r! hhhhhU hK#hU]os.chdir("/Volumes/HOME/talks/2006-08-28-musa/sage_notebook/worksheets/_scratch_/cells/35")
vr" hU+sage_notebook/worksheets/_scratch_/cells/35r# hTx [6.3626138940, 8.6035396196, 10.035509098, 11.451258611, 13.568639059, 15.914072603, 17.033610322, 17.941433571, 19.185724974, 20.379260466, 22.172490291, 23.301415503, 25.209868424, 25.876403078, 27.067635231, 28.449649699, 28.683909886, 29.974859945, 31.663575567, 33.082842812, 34.112852469, 35.236487791, 35.722877815, 37.036405161, 37.761957169, 38.833486363, 40.146656170, 41.037794024, 42.511732787, 43.429098606, 44.746454522, 45.158714131, 45.910261959, 46.757598966, 47.972050622, 48.807518125, 50.243942186, 51.371365905, 52.256372154, 52.913351819, 54.007412255, 54.763025627, 55.571029574, 56.479776666, 57.331691325, 58.183661923, 59.560440168, 60.612493128, 61.613971397, 62.338740170, 63.026897565, 63.897679433, 64.640660584, 65.171858370, 66.398517817, 67.681926727, 68.283403933, 69.303957015, 70.384072810, 71.178262591, 71.996895909, 72.547055244, 73.572601587, 74.415609866, 74.609621376, 75.881301075, 77.126021892, 77.677088797, 79.276860297, 79.804298520, 80.362782776, 81.052198857, 81.952289850, 82.649395257, 83.562455326, 84.219228268, 85.170387894, 86.241674036, 86.983793080, 88.016290486, 88.824549288, 89.626825452, 90.451337993, 91.004403919, 91.797839612, 92.165623993, 93.192461342, 94.123787433, 95.081284195, 96.168552220, 96.971843839, 97.590110540, 98.382381290, 99.228058964, 100.00116637, 100.21146715, 101.49875748, 101.87562618, 103.10778916, 103.52390456]r$ h h!h"h$h%jV h'h(ub(hor% }r& (hU hhhhU hK$hU+sage_notebook/worksheets/_scratch_/cells/36r' hU h h%jV h(ubeU_Worksheet__synchror( MJU_Worksheet__comp_is_runningr) U_Worksheet__dirr* U"sage_notebook/worksheets/_scratch_r+ U_Worksheet__attachedr, }r- U_Worksheet__queuer. ]r/ U_Worksheet__next_idr0 K.U_Worksheet__namer1 U _scratch_r2 U_Worksheet__notebookr3 hU_Worksheet__idr4 K U_Worksheet__next_block_idr5 K U_Worksheet__systemr6 NubuU_Notebook__historyr7 ]r8 (U~# Worksheet '_scratch_' (2006-08-17 at 15:14)
sage: view(f)
x^{389} + \frac{17}{3}x + 2r9 T # Worksheet '_scratch_' (2006-08-17 at 15:14)
hide%html> You can embed HTML
hide%html> It can even include math: \prod(1-q^n)
You can embed HTML
It can even include math: \prod(1-q^n)r: U# Worksheet '_scratch_' (2006-08-17 at 15:14)
latex> Cells can be written in latex, which can refer to SAGE objects.
latex> For example, consider $E$ given by $\sage{E}$.
r; U# Worksheet '_scratch_' (2006-08-17 at 15:14)
time> f = x^389 + 17/3*x + 2
time> g = x^397 - 18*x + 15
time> h = f*g^10
CPU time: 0.01 s, Wall time: 0.01 sr< U# Worksheet '_scratch_' (2006-08-17 at 15:14)
time> f = x^389 + 17/3*x + 2
time> g = x^397 - 18*x + 15
time> h = f*g^10 + f*g
CPU time: 0.01 s, Wall time: 0.01 sr= U# Worksheet '_scratch_' (2006-08-17 at 15:14)
time> f = x^389 + 17/3*x + 2
time> g = x^397 - 18*x + 15
time> h = f*g^10 + f^5*g
CPU time: 0.01 s, Wall time: 0.01 sr> U# Worksheet '_scratch_' (2006-08-17 at 15:14)
time> f = x^389 + 17/3*x + 2
time> g = x^397 - 18*x + 15
time> h = f*g^10 + f^10*g
CPU time: 0.02 s, Wall time: 0.02 sr? U# Worksheet '_scratch_' (2006-08-17 at 15:15)
time> f = x^389 + 17/3*x + 2
time> g = x^397 - 18*x + 15
time> h = f*g^10 + f^10*g
CPU time: 0.01 s, Wall time: 0.01 sr@ TL # Worksheet '_scratch_' (2006-08-17 at 15:15)
sage: h[:50]
[1153300796610, -10571923411357, 35521669089060, -82207173890380/3, -1578378351911680/9, 6297739398214880/9, -107392893078774688/81, 377409713492725760/243, -287199547816076060/243, 3757883697935134460/6561, -3178137624289098691/19683, 132742369030852798/6561, 0, 0, 0, ...rA U9# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: 2^3
8rB U;# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: 2/3
2/3rC UX# Worksheet '_scratch_' (2006-08-17 at 17:00)
python> print(2^3)
python> print(2/3)
1
0rD UJ# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: factor(2007)
3^2 * 223rE U# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: M = MatrixSpace(QQ,3)([1,2,3,1/3,17,-2/3,1,5,-5])
sage: M
[ 1 2 3]
[ 1/3 17 -2/3]
[ 1 5 -5]rF U|# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: M.parent()
Full MatrixSpace of 3 by 3 dense matrices over Rational FieldrG U# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: view(M)
\left(\begin{array}{rrr}
1&2&3\\
\frac{1}{3}&17&-\frac{2}{3}\\
1&5&-5
\end{array}\right)rH U^# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: M.echelon_form()
[1 0 0]
[0 1 0]
[0 0 1]rI UL# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: G = AlternatingGroup(5)
rJ U# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: G.conjugacy_classes_representatives()
[(), (1,2)(3,4), (1,2,3), (1,2,3,4,5), (1,2,3,5,4)]rK U# Worksheet '_scratch_' (2006-08-17 at 17:00)
time> f = x^389 + 17/3*x + 2
time> g = x^397 - 18*x + 15
time> h = f*g^10 + f^10*g
CPU time: 0.01 s, Wall time: 0.01 srL TL # Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: h[:50]
[1153300796610, -10571923411357, 35521669089060, -82207173890380/3, -1578378351911680/9, 6297739398214880/9, -107392893078774688/81, 377409713492725760/243, -287199547816076060/243, 3757883697935134460/6561, -3178137624289098691/19683, 132742369030852798/6561, 0, 0, 0, ...rM U~# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: view(f)
x^{389} + \frac{17}{3}x + 2rN T # Worksheet '_scratch_' (2006-08-17 at 17:00)
hide%html> You can embed HTML
hide%html> It can even include math: \prod(1-q^n)
You can embed HTML
It can even include math: \prod(1-q^n)rO U# Worksheet '_scratch_' (2006-08-17 at 17:00)
latex> Cells can be written in latex, which can refer to SAGE objects.
latex> For example, consider $E$ given by $\sage{E}$.
rP Uo# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: time n=factorial(10^6)
CPU time: 3.25 s, Wall time: 3.28 srQ U# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: f = maxima('x*sin(x)*cos(x)^2')
sage: print f, type(f)
x*cos(x)^2*sin(x) rR Uv# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: f.integrate()
(sin(3*x) - 3*x*cos(3*x) + 9*sin(x) - 9*x*cos(x))/36rS UD# Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: f.name()
'sage0'rT T # Worksheet '_scratch_' (2006-08-17 at 17:00)
# Many random spheres:
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: k=0
sage: for i in range(100):
... k += 1
... t.sphere((random(),random(), random()), random()/10, 't%s'%(k%3))
...
sage: t.save() # long (several seconds)
rU T # Worksheet '_scratch_' (2006-08-17 at 17:00)
sage: plot3dsoya(lambda x,y: abs(zeta(x+I*y)),(1,0), side=4, res=32).show()
[> ] 0%[----------> ] 17%[--------------------> ] 34%[------------------------------------> ] 60%[--...rV U9# Worksheet '_scratch_' (2006-08-17 at 17:23)
sage: 2^3
8rW U;# Worksheet '_scratch_' (2006-08-17 at 17:23)
sage: 2/3
2/3rX UX# Worksheet '_scratch_' (2006-08-17 at 17:24)
python> print(2^3)
python> print(2/3)
1
0rY UJ# Worksheet '_scratch_' (2006-08-17 at 17:24)
sage: factor(2007)
3^2 * 223rZ U# Worksheet '_scratch_' (2006-08-17 at 17:25)
sage: M = MatrixSpace(QQ,3)([1,2,3,1/3,17,-2/3,1,5,-5])
sage: M
[ 1 2 3]
[ 1/3 17 -2/3]
[ 1 5 -5]r[ U|# Worksheet '_scratch_' (2006-08-17 at 17:25)
sage: M.parent()
Full MatrixSpace of 3 by 3 dense matrices over Rational Fieldr\ U# Worksheet '_scratch_' (2006-08-17 at 17:25)
sage: view(M)
\left(\begin{array}{rrr}
1&2&3\\
\frac{1}{3}&17&-\frac{2}{3}\\
1&5&-5
\end{array}\right)r] U^# Worksheet '_scratch_' (2006-08-17 at 17:27)
sage: M.echelon_form()
[1 0 0]
[0 1 0]
[0 0 1]r^ U# Worksheet '_scratch_' (2006-08-17 at 17:27)
sage: view(M.echelon_form())
\left(\begin{array}{rrr}
1&0&0\\
0&1&0\\
0&0&1
\end{array}\right)r_ UL# Worksheet '_scratch_' (2006-08-17 at 17:27)
sage: G = AlternatingGroup(5)
r` U# Worksheet '_scratch_' (2006-08-17 at 17:27)
sage: G.conjugacy_classes_representatives()
[(), (1,2)(3,4), (1,2,3), (1,2,3,4,5), (1,2,3,5,4)]ra U# Worksheet '_scratch_' (2006-08-17 at 17:28)
time> f = x^389 + 17/3*x + 2
time> g = x^397 - 18*x + 15
time> h = f*g^10 + f^10*g
CPU time: 0.02 s, Wall time: 0.02 srb TL # Worksheet '_scratch_' (2006-08-17 at 17:28)
sage: h[:50]
[1153300796610, -10571923411357, 35521669089060, -82207173890380/3, -1578378351911680/9, 6297739398214880/9, -107392893078774688/81, 377409713492725760/243, -287199547816076060/243, 3757883697935134460/6561, -3178137624289098691/19683, 132742369030852798/6561, 0, 0, 0, ...rc U~# Worksheet '_scratch_' (2006-08-17 at 17:28)
sage: view(f)
x^{389} + \frac{17}{3}x + 2rd T # Worksheet '_scratch_' (2006-08-17 at 17:28)
hide%html> You can embed HTML
hide%html> It can even include math: \prod(1-q^n)
You can embed HTML
It can even include math: \prod(1-q^n)re U# Worksheet '_scratch_' (2006-08-17 at 17:29)
latex> Cells can be written in latex, which can refer to SAGE objects.
latex> For example, consider $E$ given by $\sage{E}$.
rf Uo# Worksheet '_scratch_' (2006-08-17 at 17:31)
sage: time n=factorial(10^6)
CPU time: 3.31 s, Wall time: 3.34 srg U# Worksheet '_scratch_' (2006-08-17 at 17:31)
sage: f = maxima('x*sin(x)*cos(x)^2')
sage: print f, type(f)
x*cos(x)^2*sin(x) rh Uv# Worksheet '_scratch_' (2006-08-17 at 17:32)
sage: f.integrate()
(sin(3*x) - 3*x*cos(3*x) + 9*sin(x) - 9*x*cos(x))/36ri T # Worksheet '_scratch_' (2006-08-17 at 17:32)
# Many random spheres:
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: k=0
sage: for i in range(100):
... k += 1
... t.sphere((random(),random(), random()), random()/10, 't%s'%(k%3))
...
sage: t.save() # long (several seconds)
rj U# Worksheet '_scratch_' (2006-08-27 at 04:21)
sage: M = MatrixSpace(QQ,3)([1,2,3,1/3,17,-2/3,1,5,-5])
sage: M
[ 1 2 3]
[ 1/3 17 -2/3]
[ 1 5 -5]rk U|# Worksheet '_scratch_' (2006-08-27 at 04:21)
sage: M.parent()
Full MatrixSpace of 3 by 3 dense matrices over Rational Fieldrl U# Worksheet '_scratch_' (2006-08-27 at 04:21)
sage: view(M)
\left(\begin{array}{rrr}
1&2&3\\
\frac{1}{3}&17&-\frac{2}{3}\\
1&5&-5
\end{array}\right)rm U9# Worksheet '_scratch_' (2006-08-27 at 04:21)
sage: 2^3
8rn U;# Worksheet '_scratch_' (2006-08-27 at 04:21)
sage: 2/3
2/3ro UX# Worksheet '_scratch_' (2006-08-27 at 04:21)
python> print(2^3)
python> print(2/3)
1
0rp UJ# Worksheet '_scratch_' (2006-08-27 at 04:21)
sage: factor(2007)
3^2 * 223rq U|# Worksheet '_scratch_' (2006-08-27 at 04:21)
sage: M.parent()
Full MatrixSpace of 3 by 3 dense matrices over Rational Fieldrr U# Worksheet '_scratch_' (2006-08-27 at 04:21)
sage: view(M)
\left(\begin{array}{rrr}
1&2&3\\
\frac{1}{3}&17&-\frac{2}{3}\\
1&5&-5
\end{array}\right)rs U# Worksheet '_scratch_' (2006-08-27 at 04:23)
sage: view(M.echelon_form())
\left(\begin{array}{rrr}
1&0&0\\
0&1&0\\
0&0&1
\end{array}\right)rt Up# Worksheet '_scratch_' (2006-08-27 at 04:23)
sage: view(2/3)
\frac{2}{3}ru U# Worksheet '_scratch_' (2006-08-27 at 04:23)
sage: view(M)
\left(\begin{array}{rrr}
1&2&3\\
\frac{1}{3}&17&-\frac{2}{3}\\
1&5&-5
\end{array}\right)rv T # Worksheet '_scratch_' (2006-08-27 at 04:29)
sage: # Many random spheres:
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-10,0),(0,1,0),'white')
...
...
sage: k=0
sage: for i in range(100):
... k += 1
... t.sphere((random(),random(), random()), random()/10, 't%s'%(k%3))
...
...
sage: t.save()
rw T # Worksheet '_scratch_' (2006-08-27 at 04:30)
sage: # Many random spheres:
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-10,0),(.1,1,.1),'white')
...
...
sage: k=0
sage: for i in range(100):
... k += 1
... t.sphere((random(),random(), random()), random()/10, 't%s'%(k%3))
...
...
sage: t.save()
rx T # Worksheet '_scratch_' (2006-08-27 at 04:30)
sage: # Many random spheres:
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-10,0),(0,1,.1),'white')
...
...
sage: k=0
sage: for i in range(100):
... k += 1
... t.sphere((random(),random(), random()), random()/10, 't%s'%(k%3))
...
...
sage: t.save()
ry T # Worksheet '_scratch_' (2006-08-27 at 04:30)
sage: # Many random spheres:
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-10,0),(0,1,0),'white')
...
...
sage: k=0
sage: for i in range(100):
... k += 1
... t.sphere((random(),random(), random()), random()/10, 't%s'%(k%3))
...
...
sage: t.save()
rz T # Worksheet '_scratch_' (2006-08-27 at 04:30)
sage: # Many random spheres:
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100,0),(0,1,0),'white')
sage: t.plane((-100,0,0),(1,0,0),'white')
...
...
sage: k=0
sage: for i in range(100):
... k += 1
... t.sphere((random(),random(), random()), random()/10, 't%s'%(k%3))
...
...
sage: t.save()
r{ T # Worksheet '_scratch_' (2006-08-27 at 04:31)
sage: # Many random spheres:
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100,0),(0,1,0),'white')
sage: t.plane((-100,0,0),(1,0,0),'white')
...
...
sage: k=0
sage: for i in range(100):
... k += 1
... t.sphere((random(),random(), random()), random()/10, 't%s'%(k%3))
...
...
sage: t.save()
r| T5 # Worksheet '_scratch_' (2006-08-27 at 04:32)
sage: # Many random spheres:
sage: t = Tachyon(xres=512,yres=512, camera_center=(2,0.5,0.5), look_at=(0.5,0.5,0.5), raydepth=4)
sage: t.light((4,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100,0),(0,1,0),'white')
sage: t.plane((-100,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r} T
# Worksheet '_scratch_' (2006-08-27 at 04:32)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(5,2,2), look_at=(0,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100,0),(0,1,0),'white')
sage: t.plane((-100,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r~ T # Worksheet '_scratch_' (2006-08-27 at 04:32)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(5,2,2), look_at=(0,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-1000,0),(0,1,0),'white')
sage: t.plane((-1000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:33)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(5,2,2), look_at=(0,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:33)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(5,2,2), look_at=(0,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:33)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(5,2,2), look_at=(-.5,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:33)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(5,2,4), look_at=(-.5,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:33)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(5,4,3), look_at=(-.5,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:33)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(5,4,4), look_at=(-.5,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:34)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(5,6,5), look_at=(-.5,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:34)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(8,6,5), look_at=(-.5,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:34)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(8,5,7), look_at=(-.5,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], ZZ(i)/n), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:34)
sage: # Many random spheres:
sage: t = Tachyon(camera_center=(8,5,7), look_at=(-.5,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], 0), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T$ # Worksheet '_scratch_' (2006-08-27 at 04:35)
sage: # Many random spheres:
sage: t = Tachyon(xres=1000, yres=800, camera_center=(8,5,7), look_at=(-.5,1,0))
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], 0), 0.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:36)
sage: # Many random spheres:
sage: t = Tachyon(xres=1000, yres=800, camera_center=(8,5,7), look_at=(-.5,1,0), ray_depth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], 0), i/(10*n)+.1, 't%s'%(i%3))
...
...
sage: t.save()
Traceback (most recent call last):
...
TypeError: __init__() got an unexpected keyword argument 'ray_depth'r T # Worksheet '_scratch_' (2006-08-27 at 04:36)
sage: # Many random spheres:
sage: t = Tachyon(xres=1000, yres=800, camera_center=(8,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... t.sphere((Q[1], Q[0], 0), i/(10*n)+.1, 't%s'%(i%3))
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:38)
sage: # Many random spheres:
sage: t = Tachyon(xres=1000, yres=800, camera_center=(8,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,-100000,0),(0,1,0),'white')
sage: t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:39)
sage: # Many random spheres:
sage: t = Tachyon(xres=1000, yres=800, camera_center=(8,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),'black')
sage: t.cylinder((0,0,0),(0,1,0),'black')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
Traceback (most recent call last):
...
TypeError: cylinder() takes exactly 5 arguments (4 given)r Tm # Worksheet '_scratch_' (2006-08-27 at 04:39)
sage: # Many random spheres:
sage: t = Tachyon(xres=1000, yres=800, camera_center=(8,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.1,'black')
sage: t.cylinder((0,0,0),(0,1,0),.1,'black')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r Tn # Worksheet '_scratch_' (2006-08-27 at 04:40)
sage: # Many random spheres:
sage: t = Tachyon(xres=1000, yres=800, camera_center=(8,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.1,'black')
sage: t.cylinder((0,0,0),(0,1,0),.1,'black')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r Tn # Worksheet '_scratch_' (2006-08-27 at 04:40)
sage: # Many random spheres:
sage: t = Tachyon(xres=1000, yres=800, camera_center=(0,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.1,'black')
sage: t.cylinder((0,0,0),(0,1,0),.1,'black')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T # Worksheet '_scratch_' (2006-08-27 at 04:41)
sage: # Many random spheres:
sage: t = Tachyon(xres=1000, yres=800, camera_center=(0,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.1,'black')
sage: t.cylinder((0,0,0),(0,1,0),.1,'black')
sage: E = EllipticCurve('37a')
sage: show(E)
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
y^2 + y = x^3 - x
r T # Worksheet '_scratch_' (2006-08-27 at 04:41)
sage: # Many random spheres:
sage: t = Tachyon(xres=1000, yres=800, camera_center=(0,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.1,'black')
sage: t.cylinder((0,0,0),(0,1,0),.1,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r Ua# Worksheet '_scratch_' (2006-08-27 at 04:41)
sage: E = EllipticCurve('37a')
sage: show(plot(E))
r Us# Worksheet '_scratch_' (2006-08-27 at 04:41)
sage: E = EllipticCurve('37a')
sage: show(plot(E, rgbcolor=(1,0,0)))
r U# Worksheet '_scratch_' (2006-08-27 at 04:42)
sage: E = EllipticCurve('37a')
sage: show(plot(E, rgbcolor=(1,0,0)))
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Fieldr Ug# Worksheet 'bsd' (2006-08-27 at 04:42)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: #y^2 + y = x^3 - x
r U# Worksheet 'bsd' (2006-08-27 at 04:43)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: #y^2 + y = x^3 - x
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Fieldr U# Worksheet 'bsd' (2006-08-27 at 04:43)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: print E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Fieldr U)# Worksheet 'bsd' (2006-08-27 at 04:43)
r U# Worksheet 'bsd' (2006-08-27 at 04:43)
html> The Birch and Swinnerton-Dyer Conjecture for 37A
The Birch and Swinnerton-Dyer Conjecture for 37A
r U# Worksheet 'bsd' (2006-08-27 at 04:44)
hide%html> The Birch and Swinnerton-Dyer Conjecture for 37A
The Birch and Swinnerton-Dyer Conjecture for 37A
r U# Worksheet 'bsd' (2006-08-27 at 04:44)
sage: P = plot(E, rgbcolor=(1,0,0))
sage: P
Graphics object consisting of 2 graphics primitivesr U# Worksheet 'bsd' (2006-08-27 at 04:44)
sage: P = plot(E, rgbcolor=(1,0,0), thickness=2)
sage: P
Graphics object consisting of 2 graphics primitivesr U6# Worksheet 'bsd' (2006-08-27 at 04:44)
sage: show(P)
r U6# Worksheet 'bsd' (2006-08-27 at 04:44)
sage: show(P)
r Te # Worksheet '_scratch_' (2006-08-27 at 04:44)
sage: t = Tachyon(xres=1000, yres=800, camera_center=(0,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.1,'black')
sage: t.cylinder((0,0,0),(0,1,0),.1,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T^ # Worksheet 'bsd' (2006-08-27 at 04:44)
sage: t = Tachyon(xres=200, yres=200, camera_center=(0,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.1,'black')
sage: t.cylinder((0,0,0),(0,1,0),.1,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T_ # Worksheet 'bsd' (2006-08-27 at 04:45)
sage: t = Tachyon(xres=200, yres=200, camera_center=(-8,5,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.1,'black')
sage: t.cylinder((0,0,0),(0,1,0),.1,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T^ # Worksheet 'bsd' (2006-08-27 at 04:45)
sage: t = Tachyon(xres=200, yres=200, camera_center=(8,2,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.1,'black')
sage: t.cylinder((0,0,0),(0,1,0),.1,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T` # Worksheet 'bsd' (2006-08-27 at 04:45)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,2,7), look_at=(-.5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T` # Worksheet 'bsd' (2006-08-27 at 04:45)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,2,7), look_at=(-.2,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T_ # Worksheet 'bsd' (2006-08-27 at 04:46)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,2,7), look_at=(-2,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T_ # Worksheet 'bsd' (2006-08-27 at 04:46)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,2,7), look_at=(-5,1,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T_ # Worksheet 'bsd' (2006-08-27 at 04:46)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,2,7), look_at=(-1,3,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T_ # Worksheet 'bsd' (2006-08-27 at 04:46)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,2,7), look_at=(-1,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T` # Worksheet 'bsd' (2006-08-27 at 04:46)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,2,7), look_at=(-10,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[1], -Q[0], 0), c, 't%s'%i)
...
...
sage: t.save()
r T^ # Worksheet 'bsd' (2006-08-27 at 04:46)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,2,7), look_at=(0,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[0], -Q[1], 0), c, 't%s'%i)
...
...
sage: t.save()
r T] # Worksheet 'bsd' (2006-08-27 at 04:47)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,2,7), look_at=(0,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[0], Q[1], 0), c, 't%s'%i)
...
...
sage: t.save()
r T^ # Worksheet 'bsd' (2006-08-27 at 04:47)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,10,7), look_at=(0,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[0], Q[1], 0), c, 't%s'%i)
...
...
sage: t.save()
r T^ # Worksheet 'bsd' (2006-08-27 at 04:47)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,10,3), look_at=(0,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[0], Q[1], 0), c, 't%s'%i)
...
...
sage: t.save()
r T^ # Worksheet 'bsd' (2006-08-27 at 04:47)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,10,5), look_at=(0,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[0], Q[1], 0), c, 't%s'%i)
...
...
sage: t.save()
r T^ # Worksheet 'bsd' (2006-08-27 at 04:47)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/(10*n) + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[0], Q[1], 0), c, 't%s'%i)
...
...
sage: t.save()
r T\ # Worksheet 'bsd' (2006-08-27 at 04:48)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(1.0,c,c))
... t.sphere((Q[0], Q[1], 0), c/10, 't%s'%i)
...
...
sage: t.save()
r TZ # Worksheet 'bsd' (2006-08-27 at 04:48)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(0,c,c))
... t.sphere((Q[0], Q[1], 0), c/10, 't%s'%i)
...
...
sage: t.save()
r TZ # Worksheet 'bsd' (2006-08-27 at 04:48)
sage: t = Tachyon(xres=200, yres=200, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(c,0,c))
... t.sphere((Q[0], Q[1], 0), c/10, 't%s'%i)
...
...
sage: t.save()
r TZ # Worksheet 'bsd' (2006-08-27 at 04:48)
sage: t = Tachyon(xres=500, yres=500, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(c,0,c))
... t.sphere((Q[0], Q[1], 0), c/10, 't%s'%i)
...
...
sage: t.save()
r TZ # Worksheet 'bsd' (2006-08-27 at 04:49)
sage: t = Tachyon(xres=500, yres=500, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 0.2, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(1,1,1))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 400
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(c,0,c))
... t.sphere((Q[0], Q[1], 0), c/10, 't%s'%i)
...
...
sage: t.save()
r TX # Worksheet 'bsd' (2006-08-27 at 04:50)
sage: t = Tachyon(xres=500, yres=500, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.05,'black')
sage: t.cylinder((0,0,0),(0,1,0),.05,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 400
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(c,0,c))
... t.sphere((Q[0], Q[1], 0), c/10, 't%s'%i)
...
...
sage: t.save()
r T} # Worksheet 'bsd' (2006-08-27 at 04:51)
sage: t = Tachyon(xres=500, yres=500, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
sage: t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
sage: t.texture('white', color=(1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: #t.plane((0,-100000,0),(0,1,0),'white')
sage: #t.plane((-100000,0,0),(1,0,0),'white')
sage: t.plane((-100000,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... #t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(c,0,c))
... t.sphere((Q[0], Q[1], 0), c/10, 'red')
...
...
sage: t.save()
r U# Worksheet 'bsd' (2006-08-27 at 04:51)
sage: G = plot(E, rgbcolor=(1,0,0), thickness=2)
sage: G
Graphics object consisting of 2 graphics primitivesr U6# Worksheet 'bsd' (2006-08-27 at 04:51)
sage: show(G)
r UI# Worksheet 'bsd' (2006-08-27 at 04:51)
sage: P = E([0,0]); P
(0 : 0 : 1)r U=# Worksheet 'bsd' (2006-08-27 at 04:51)
sage: 2*P
(1 : 0 : 1)r U}# Worksheet 'bsd' (2006-08-27 at 04:52)
sage: 2*P, 3*P, 4*P, 5*P
((1 : 0 : 1), (-1 : -1 : 1), (2 : -3 : 1), (1/4 : -5/8 : 1))r UJ# Worksheet 'bsd' (2006-08-27 at 04:52)
sage: 10*P
(161/16 : -2065/64 : 1)r U\# Worksheet 'bsd' (2006-08-27 at 04:52)
sage: for n in range(-10,10):
... G += plot(n*P)
r U\# Worksheet 'bsd' (2006-08-27 at 04:52)
sage: for n in range(-10,10):
... G += plot(n*P)
r U# Worksheet 'bsd' (2006-08-27 at 04:52)
sage: GE = plot(E, rgbcolor=(1,0,0), thickness=2)
sage: GE
Graphics object consisting of 2 graphics primitivesr U7# Worksheet 'bsd' (2006-08-27 at 04:52)
sage: show(GE)
r Up# Worksheet 'bsd' (2006-08-27 at 04:53)
sage: G = GE + sum([plot(n*P,rgbcolor=(0,0,1)) for n in range(-10,10)])
r U~# Worksheet 'bsd' (2006-08-27 at 04:53)
sage: G = GE + sum([plot(n*P,rgbcolor=(0,0,1)) for n in range(-10,10)])
sage: show(G)
r U# Worksheet 'bsd' (2006-08-27 at 04:53)
sage: G = GE + sum([plot(n*P,rgbcolor=(0,0,1)) for n in range(-10,10)])
sage: show(G,xmax=2.5,xmin=-2)
r U# Worksheet 'bsd' (2006-08-27 at 04:54)
sage: G = GE + sum([plot(n*P,rgbcolor=(0,0,1)) for n in range(-10,10)])
sage: show(G,xmax=2.5,xmin=-2,ymax=5,ymin=-5)
r U# Worksheet 'bsd' (2006-08-27 at 04:55)
sage: G = Graphics()
sage: Q = P
sage: for n in range(50):
... Q = Q + P
... if abs(Q[0]) < 3 and abs(Q[1]) < 5:
... G += point(Q,rgbcolor=(0,0,1))
...
sage: show(GE+G)
r T # Worksheet 'bsd' (2006-08-27 at 04:56)
sage: G = plot(E, rgbcolor=(1,0,0), thickness=.5)
sage: Q = P
sage: for n in range(50):
... Q = Q + P
... if abs(Q[0]) < 3 and abs(Q[1]) < 5:
... G += point(Q,rgbcolor=(0,0,1),pointsize=40)
...
sage: show(G)
r T # Worksheet 'bsd' (2006-08-27 at 04:56)
sage: G = plot(E, rgbcolor=(1,1,1), thickness=.3)
sage: Q = P
sage: for n in range(50):
... Q = Q + P
... if abs(Q[0]) < 3 and abs(Q[1]) < 5:
... G += point(Q,rgbcolor=(1,0,0),pointsize=40)
...
sage: show(G)
r U# Worksheet 'bsd' (2006-08-27 at 04:56)
sage: G = plot(E, thickness=.3)
sage: Q = P
sage: for n in range(50):
... Q = Q + P
... if abs(Q[0]) < 3 and abs(Q[1]) < 5:
... G += point(Q,rgbcolor=(1,0,0),pointsize=40)
...
sage: show(G)
r U# Worksheet 'bsd' (2006-08-27 at 04:56)
sage: G = plot(E, thickness=.3)
sage: Q = P
sage: for n in range(100):
... Q = Q + P
... if abs(Q[0]) < 3 and abs(Q[1]) < 5:
... G += point(Q,rgbcolor=(1,0,0),pointsize=40)
...
sage: show(G)
r T # Worksheet 'bsd' (2006-08-27 at 04:57)
sage: t = Tachyon(xres=500, yres=500, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((-1,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... #t.texture('t%s'%i, ambient=0.1,diffuse=0.9,specular=0.5,color=(c,0,c))
... t.sphere((Q[0], Q[1], 0), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 04:58)
sage: t = Tachyon(xres=500, yres=500, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.sphere((Q[0], Q[1], 0), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 04:58)
sage: t = Tachyon(xres=500, yres=500, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,-1),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.sphere((Q[0], Q[1], 0), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 04:58)
sage: t = Tachyon(xres=500, yres=500, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,-1),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.sphere((Q[0], -Q[1], 0), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 04:59)
sage: t = Tachyon(xres=500, yres=500, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,-.1),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.sphere((Q[0], -Q[1], 0), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 04:59)
sage: t = Tachyon(xres=500, yres=500, camera_center=(4,10,5), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.sphere((Q[0], -Q[1], c/10), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 04:59)
sage: t = Tachyon(xres=500, yres=500, camera_center=(3,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.sphere((Q[0], -Q[1], c/10), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:00)
sage: G = plot(E, thickness=.3)
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... if abs(Q[0]) < 3 and abs(Q[1]) < 5:
... G += point(Q,rgbcolor=(1,0,0),pointsize=10+i/n)
...
sage: show(G)
r T # Worksheet 'bsd' (2006-08-27 at 05:00)
sage: G = plot(E, thickness=.3)
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... if abs(Q[0]) < 3 and abs(Q[1]) < 5:
... G += point(Q,rgbcolor=(1,0,0),pointsize=10+float(i)*100/n)
...
sage: show(G)
r U# Worksheet 'bsd' (2006-08-27 at 05:01)
sage: 30*P
(79799551268268089761/62586636021357187216 : -754388827236735824355996347601/495133617181351428873673516736 : 1)r T # Worksheet 'bsd' (2006-08-27 at 05:01)
sage: t = Tachyon(xres=500, yres=500, camera_center=(3,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.sphere((Q[0], -Q[1], c/10), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:01)
sage: t = Tachyon(xres=500, yres=500, camera_center=(3,4,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.sphere((Q[0], -Q[1], c/10), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:01)
sage: t = Tachyon(xres=500, yres=500, camera_center=(1,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.sphere((Q[0], -Q[1], c/10), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:01)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.sphere((Q[0], -Q[1], c/10), c/10, 'red')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:02)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.text('r',color=(1,c,c))
... t.sphere((Q[0], -Q[1], .2), c/10, 'r')
...
...
sage: t.save()
Traceback (most recent call last):
...
AttributeError: 'Tachyon' object has no attribute 'text'r T # Worksheet 'bsd' (2006-08-27 at 05:03)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r',color=(1,c,c))
... t.sphere((Q[0], -Q[1], .2), c/10, 'r')
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:03)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(1,i/n,i/n))
... t.sphere((Q[0], -Q[1], .2), c/10, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:03)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(i/n*2,0,0))
... t.sphere((Q[0], -Q[1], .2), c/10, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:03)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(float(i)/n,0,0))
... t.sphere((Q[0], -Q[1], .2), c/10, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:04)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(float(i/n),0,0))
... t.sphere((Q[0], -Q[1], .2), c/10, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:04)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(float(i/n),0,0))
... t.sphere((Q[0], -Q[1], .01), .1, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:04)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(float(i/n),0,0))
... t.sphere((Q[0], -Q[1], .01), .02, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:05)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 200
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(float(i/n),0,0))
... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:05)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(float(i/n),0,0))
... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:05)
sage: t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(float(i/n),0,0))
... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:06)
sage: t = Tachyon(xres=800, yres=600, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: show(plot(E))
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(float(i/n),0,0))
... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:06)
sage: t = Tachyon(xres=800, yres=600, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... c = i/n + .1
... t.texture('r%s'%i,color=(float(i/n),0,0))
... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i)
...
...
sage: t.save()
r U# Worksheet 'bsd' (2006-08-27 at 05:06)
hide%html> In sharp contrast, the curve
In sharp contrast, the curver U# Worksheet 'bsd' (2006-08-27 at 05:07)
sage: EC('11a')
Traceback (most recent call last):
...
NameError: name 'EC' is not definedr U# Worksheet 'bsd' (2006-08-27 at 05:07)
sage: EllipticCurve('11a')
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Fieldr Ti # Worksheet 'bsd' (2006-08-27 at 05:07)
hide%html> In sharp contrast, the curve
hide%html> y^2 + y = x^3 - x^2 - 10*x - 20
hide%html> has only finitely many rational points!
In sharp contrast, the curve
y^2 + y = x^3 - x^2 - 10*x - 20
has only finitely many rational points!
r To # Worksheet 'bsd' (2006-08-27 at 05:08)
hide%html> In sharp contrast, the following curve has only finitely many rational points!
hide%html> y^2 + y = x^3 - x^2 - 10*x - 20
In sharp contrast, the following curve has only finitely many rational points!
y^2 + y = x^3 - x^2 - 10*x - 20r Tm # Worksheet 'bsd' (2006-08-27 at 05:08)
hide%html> In sharp contrast, the following curve has only finitely many rational points!
hide%html> y^2 + y = x^3 - x^2 - 10x - 20
In sharp contrast, the following curve has only finitely many rational points!
y^2 + y = x^3 - x^2 - 10x - 20r T # Worksheet 'bsd' (2006-08-27 at 05:08)
hide%html> In sharp contrast, the following curve has only finitely many rational points!
hide%html>
hide%html> y^2 + y = x^3 - x^2 - 10x - 20
hide%html>
In sharp contrast, the following curve has only finitely many rational points!
y^2 + y = x^3 - x^2 - 10x - 20
r T # Worksheet 'bsd' (2006-08-27 at 05:08)
hide%html> In sharp contrast, the following curve has only finitely many rational points!
hide%html>
hide%html> y^2 + y = x^3 - x^2 - 10x - 20
hide%html>
In sharp contrast, the following curve has only finitely many rational points!
y^2 + y = x^3 - x^2 - 10x - 20
r T # Worksheet 'bsd' (2006-08-27 at 05:09)
hide%html> In sharp contrast, the following curve has only finitely many rational points!
hide%html>
hide%html> y^2 + y = x^3 - x^2 - 10x - 20
hide%html>
hide%html> Question: Is there an a priori way to tell which type
hide%html> of elliptic curve we
In sharp contrast, the following curve has only finitely many rational points!
y^2 + y = x^3 - x^2 - 10x - 20
Question: Is there an a priori way to tell which type
of elliptic curve wer T # Worksheet 'bsd' (2006-08-27 at 05:09)
hide%html> In sharp contrast, the following curve has only finitely many rational points!
hide%html>
hide%html> y^2 + y = x^3 - x^2 - 10x - 20
hide%html>
hide%html> Question: Is there an a priori way to tell which type
hide%html> of elliptic curve we are dealing with?
hide%html> ..
In sharp contrast, the following curve has only finitely many rational points!
y^2 + y = x^3 - x^2 - 10x - 20
Question: Is there an a priori way to tell which type
of elliptic curve we are dealing with?
..r T # Worksheet 'bsd' (2006-08-27 at 05:09)
hide%html> In sharp contrast, the following curve has only finitely many rational points!
hide%html>
hide%html> y^2 + y = x^3 - x^2 - 10x - 20
hide%html>
hide%html> Question: Is there an a priori way to tell which type
hide%html> of elliptic curve we are dealing with?
hide%html> .
In sharp contrast, the following curve has only finitely many rational points!
y^2 + y = x^3 - x^2 - 10x - 20
Question: Is there an a priori way to tell which type
of elliptic curve we are dealing with?
.r T # Worksheet 'bsd' (2006-08-27 at 05:10)
hide%html> In sharp contrast, the following curve has only finitely many rational points!
hide%html>
hide%html> y^2 + y = x^3 - x^2 - 10x - 20
hide%html>
hide%html> Question: Is there an a priori way to tell which type
hide%html> of elliptic curve we are dealing with?
In sharp contrast, the following curve has only finitely many rational points!
y^2 + y = x^3 - x^2 - 10x - 20
Question: Is there an a priori way to tell which type
of elliptic curve we are dealing with?r T # Worksheet 'bsd' (2006-08-27 at 05:11)
hide%html> The L-function of an elliptic curve is an analytic (or holomorphic)
hide%html> function on C defined by counting points modulo primes.
hide%html>
The L-function of an elliptic curve is an analytic (or holomorphic)
function on C defined by counting points modulo primes.
r U# Worksheet 'bsd' (2006-08-27 at 05:12)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: plot(E.change_ring(GF(5)))
Graphics object consisting of 7 graphics primitivesr Uu# Worksheet 'bsd' (2006-08-27 at 05:12)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: show(plot(E.change_ring(GF(5))))
r U# Worksheet 'bsd' (2006-08-27 at 05:12)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: G = [plot(E.change_ring(GF(p)), pointsize=40, rgbcolor=(1,0,0)) for p in primes(35)]
r U# Worksheet 'bsd' (2006-08-27 at 05:13)
sage: len(primes(35))
Traceback (most recent call last):
...
TypeError: len() of unsized objectr UF# Worksheet 'bsd' (2006-08-27 at 05:13)
sage: len(list(primes(35)))
11r U# Worksheet 'bsd' (2006-08-27 at 05:13)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: G = [plot(E.change_ring(GF(p)), pointsize=40, rgbcolor=(1,0,0)) for p in primes(35)]
sage: show(graphics_array(G,4,3))
r U# Worksheet 'bsd' (2006-08-27 at 05:13)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: G = [plot(E.change_ring(GF(p)), pointsize=40, rgbcolor=(1,0,0)) for p in primes(35)]
sage: show(graphics_array(G,4,3),fontsize=4)
r U# Worksheet 'bsd' (2006-08-27 at 05:14)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: G = [plot(E.change_ring(GF(p)), pointsize=30, rgbcolor=(1,0,0)) for p in primes(42) if p!=37]
sage: show(graphics_array(G,4,3),fontsize=4)
r U# Worksheet 'bsd' (2006-08-27 at 05:15)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: for p in primes(100):
... print '%-10s-10s'%(p,E.Np(p))
Traceback (most recent call last):
...
TypeError: not all arguments converted during string formattingr T_ # Worksheet 'bsd' (2006-08-27 at 05:15)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: for p in primes(100):
... print '%-10s%-10s'%(p,E.Np(p))
2 5
3 7
5 8
7 9
11 17
13 16
17 18
19 20
23 22
29 24
31 36
37 39
41 51
43 42
47 57
53 53
59 52
61 70
67 60
71 63
73 75 ...r T # Worksheet 'bsd' (2006-08-27 at 05:16)
sage: E = EllipticCurve([0,0,1,-1,0])
sage: print E
sage: GE = plot(E, rgbcolor=(1,0,0), thickness=2)
sage: print GE
sage: show(GE)
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
Graphics object consisting of 2 graphics primitivesr T # Worksheet 'bsd' (2006-08-27 at 05:17)
hide%html> The Birch and Swinnerton-Dyer Conjecture for 37A
hide%html>
hide%html> This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
hide%html> for the elliptic curve y^2 + y = x^3 - x,
hide%html> affectionately referred to by mathematicians as 37a.
hide%html>
The Birch and Swinnerton-Dyer Conjecture for 37A
This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
for the elliptic curve y^2 + y = x^3 - x,
affectionately referred to by mathematicians as 37a.
r T # Worksheet 'bsd' (2006-08-27 at 05:17)
hide%html> The Birch and Swinnerton-Dyer Conjecture for 37A
hide%html>
hide%html> This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
hide%html> for the elliptic curve
hide%html> y^2 + y = x^3 - x,
hide%html> affectionately referred to by mathematicians as 37a.
hide%html>
The Birch and Swinnerton-Dyer Conjecture for 37A
This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
for the elliptic curve
y^2 + y = x^3 - x,
affectionately referred to by mathematicians as 37a.
r T # Worksheet 'bsd' (2006-08-27 at 05:18)
hide%html> The Birch and Swinnerton-Dyer Conjecture for 37A
hide%html>
hide%html> This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
hide%html> for the elliptic curve
hide%html> y^2 + y = x^3 - x,
hide%html> affectionately referred to by mathematicians as 37a.
hide%html>
The Birch and Swinnerton-Dyer Conjecture for 37A
This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
for the elliptic curve
y^2 + y = x^3 - x,
affectionately referred to by mathematicians as 37a.
r T # Worksheet 'bsd' (2006-08-27 at 05:18)
hide%html> The Birch and Swinnerton-Dyer Conjecture for 37A
hide%html>
hide%html> This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
hide%html> for the elliptic curve
hide%html> y^2 + y = x^3 - x,
hide%html>
hide%html> affectionately referred to by mathematicians as 37a.
hide%html>
The Birch and Swinnerton-Dyer Conjecture for 37A
This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
for the elliptic curve
y^2 + y = x^3 - x,
affectionately referred to by mathematicians as 37a.
r T # Worksheet 'bsd' (2006-08-27 at 05:18)
hide%html> The Birch and Swinnerton-Dyer Conjecture for 37A
hide%html>
hide%html> This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
hide%html> for the elliptic curve
hide%html> y^2 + y = x^3 - x,
hide%html>
hide%html> affectionately referred to by mathematicians as 37a.
hide%html>
The Birch and Swinnerton-Dyer Conjecture for 37A
This worksheet illustrates the conjecture of Birch and Swinnerton-Dyer
for the elliptic curve
y^2 + y = x^3 - x,
affectionately referred to by mathematicians as 37a.
r T # Worksheet 'bsd' (2006-08-27 at 05:18)
sage: P = E([0,0])
sage: print 2*P, 3*P, 4*P, 5*P, 30*P
(1 : 0 : 1) (-1 : -1 : 1) (2 : -3 : 1) (1/4 : -5/8 : 1) (79799551268268089761/62586636021357187216 : -754388827236735824355996347601/495133617181351428873673516736 : 1)r T # Worksheet 'bsd' (2006-08-27 at 05:18)
sage: G = plot(E, thickness=.3)
sage: Q = P
sage: n = 100
sage: for i in range(n):
... Q = Q + P
... if abs(Q[0]) < 3 and abs(Q[1]) < 5:
... G += point(Q,rgbcolor=(1,0,0),pointsize=10+float(i)*100/n)
...
sage: show(G)
r T # Worksheet 'bsd' (2006-08-27 at 05:19)
hide%html> Sharp contrast: the following curve has only finitely many rational points!
hide%html>
hide%html> y^2 + y = x^3 - x^2 - 10x - 20
hide%html>
hide%html> Question: Is there an a priori way to tell which type
hide%html> of elliptic curve we are dealing with?
hide%html>
hide%html> Question: How often does each possibility occur? (Conjecture: 50% each.)
hide%html>
Sharp contrast: the following curve has only finitely many rational points!
y^2 + y = x^3 - x^2 - 10x - 20
Question: Is there an a priori way to tell which type
of elliptic curve we are dealing with?
Question: How often does each possibility occur? (Conjecture: 50% each.)
r T # Worksheet 'bsd' (2006-08-27 at 05:19)
hide%html> Sharp contrast: the following curve has only finitely many rational points!
hide%html>
hide%html> y^2 + y = x^3 - x^2 - 10x - 20
hide%html>
hide%html>
hide%html> Question: Is there an a priori way to tell which type
hide%html> of elliptic curve we are dealing with?
hide%html>
hide%html>
hide%html> Question: How often does each possibility occur? (Conjecture: 50% each.)
hide%html>
Sharp contrast: the following curve has only finitely many rational points!
y^2 + y = x^3 - x^2 - 10x - 20
Question: Is there an a priori way to tell which type
of elliptic curve we are dealing with?
Question: How often does each possibility occur? (Conjecture: 50% each.)
r T # Worksheet 'bsd' (2006-08-27 at 05:20)
sage: # Some Pictures of Counting Points
sage: E = EllipticCurve([0,0,1,-1,0])
sage: G = [plot(E.change_ring(GF(p)), pointsize=30, rgbcolor=(1,0,0)) for p in primes(42) if p!=37]
sage: show(graphics_array(G,4,3),fontsize=4)
r T # Worksheet 'bsd' (2006-08-27 at 05:20)
sage: # Tally up the number of points
sage: E = EllipticCurve([0,0,1,-1,0])
sage: for p in primes(200):
... print '%-10s%-10s'%(p,E.Np(p))
2 5
3 7
5 8
7 9
11 17
13 16
17 18
19 20
23 22
29 24
31 36
37 39
41 51
43 42
47 57
53 53
59 52
61 70
67 60
71 63
73 75 ...r T # Worksheet 'bsd' (2006-08-27 at 05:21)
sage: # Tally up the number of points
sage: E = EllipticCurve([0,0,1,-1,0])
sage: for p in primes(200):
... print '%10s%-10s%-10s'%('',p,E.Np(p))
2 5
3 7
5 8
7 9
11 17
13 16
17 18
19 20
23 22
29 24
31 36
37 39
41 51
43 42
47 57...r
T # Worksheet 'bsd' (2006-08-27 at 05:21)
sage: # Tally up the number of points
sage: E = EllipticCurve([0,0,1,-1,0])
sage: print '%10s%-10s%-10s'%('','p','Number of Points')
sage: for p in primes(200):
... print '%10s%-10s%-10s'%('',p,E.Np(p))
p Number of Points
2 5
3 7
5 8
7 9
11 17
13 16
17 18
19 20
23 22
29 24
31 36
37 39
41 51
43 ...r T # Worksheet 'bsd' (2006-08-27 at 05:21)
sage: # Tally up the number of points
sage: E = EllipticCurve([0,0,1,-1,0])
sage: print '.'
sage: print '%10s%-10s%-10s'%('','p','Number of Points')
sage: for p in primes(200):
... print '%10s%-10s%-10s'%('',p,E.Np(p))
.
p Number of Points
2 5
3 7
5 8
7 9
11 17
13 16
17 18
19 20
23 22
29 24
31 36
37 39
41 51
...r T # Worksheet 'bsd' (2006-08-27 at 05:21)
sage: # Tally up the number of points
sage: E = EllipticCurve([0,0,1,-1,0])
sage: print '.'*30
sage: print '%10s%-10s%-10s'%('','p','Number of Points')
sage: for p in primes(200):
... print '%10s%-10s%-10s'%('',p,E.Np(p))
..............................
p Number of Points
2 5
3 7
5 8
7 9
11 17
13 16
17 18
19 20
23 22
29 24
31 36
37 39
...r
T # Worksheet 'bsd' (2006-08-27 at 05:21)
sage: # Tally up the number of points
sage: E = EllipticCurve([0,0,1,-1,0])
sage: print '.'*40
sage: print '%10s%-10s%-10s'%('','p','Number of Points')
sage: for p in primes(200):
... print '%10s%-10s%-10s'%('',p,E.Np(p))
........................................
p Number of Points
2 5
3 7
5 8
7 9
11 17
13 16
17 18
19 20
23 22
29 24
31 36
37 39 ...r T # Worksheet 'bsd' (2006-08-27 at 05:21)
sage: # Tally up the number of points
sage: E = EllipticCurve([0,0,1,-1,0])
sage: print '.'*40
sage: print '%10s%-10s%10s'%('','p','Number of Points')
sage: for p in primes(200):
... print '%10s%-10s%-10s'%('',p,E.Np(p))
........................................
p Number of Points
2 5
3 7
5 8
7 9
11 17
13 16
17 18
19 20
23 22
29 24
31 36
37 39 ...r T # Worksheet 'bsd' (2006-08-27 at 05:22)
sage: # Tally up the number of points
sage: E = EllipticCurve([0,0,1,-1,0])
sage: print '.'*40
sage: print '%10s%-5s%10s'%('','p','Number of Points')
sage: for p in primes(200):
... print '%10s%-10s%-10s'%('',p,E.Np(p))
........................................
p Number of Points
2 5
3 7
5 8
7 9
11 17
13 16
17 18
19 20
23 22
29 24
31 36
37 39
...r T # Worksheet 'bsd' (2006-08-27 at 05:22)
sage: # Tally up the number of points
sage: E = EllipticCurve([0,0,1,-1,0])
sage: print '.'*40
sage: print '%10s%-3s%10s'%('','p','Number of Points')
sage: for p in primes(200):
... print '%10s%-10s%-10s'%('',p,E.Np(p))
........................................
p Number of Points
2 5
3 7
5 8
7 9
11 17
13 16
17 18
19 20
23 22
29 24
31 36
37 39
...r UK# Worksheet 'bsd' (2006-08-27 at 05:22)
sage: L = E.Lseries_dokchitser(10)
r U4# Worksheet 'bsd' (2006-08-27 at 05:22)
sage: L(1)
0r UC# Worksheet 'bsd' (2006-08-27 at 05:22)
sage: plot(L, -2,3).show()
r Uv# Worksheet 'bsd' (2006-08-27 at 05:23)
sage: plot(L, -2,3, rgbcolor=(0,0,1), plot_points=90, plot_division=0).show()
r U# Worksheet 'bsd' (2006-08-27 at 05:23)
sage: plot(L, -2,3, rgbcolor=(0,0,1), plot_points=90, plot_division=0, thickness=2).show()
r T # Worksheet 'bsd' (2006-08-27 at 05:25)
sage: t = Tachyon(xres=800, yres=600, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=5
sage: for i in range(n):
... x = random()*2; y = random()*2
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,y,m), .1, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:26)
sage: t = Tachyon(xres=800, yres=600, camera_center=(2,3,2), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=5
sage: for i in range(n):
... x = random()*2; y = random()*2
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,y,m), .1, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:26)
sage: t = Tachyon(xres=800, yres=600, camera_center=(2,3,2), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=5
sage: for i in range(n):
... x = random()*2; y = random()*2
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .1, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:26)
sage: t = Tachyon(xres=800, yres=600, camera_center=(2,3,2), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=50
sage: for i in range(n):
... x = random()*2; y = random()*2
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .1, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:27)
sage: t = Tachyon(xres=800, yres=600, camera_center=(3,4,2), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=100
sage: for i in range(n):
... x = random()*2; y = random()*2
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .02, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:28)
sage: t = Tachyon(xres=800, yres=600, camera_center=(3,4,2), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=100
sage: for i in range(n):
... x = random()/10+1; y = random()/10
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .02, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:28)
sage: t = Tachyon(xres=800, yres=600, camera_center=(5,-1,2), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=100
sage: for i in range(n):
... x = random()/4+1; y = random()/4
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .02, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:29)
sage: t = Tachyon(xres=800, yres=600, camera_center=(3,1,2), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=100
sage: for i in range(n):
... x = random()/2+.8; y = random()/2
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .02, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:29)
sage: t = Tachyon(xres=800, yres=600, camera_center=(2,.5,1), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=20
sage: for i in range(n):
... x = random()/2+.8; y = random()/2 - .25
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .02, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:30)
sage: t = Tachyon(xres=800, yres=600, camera_center=(1.5,.4,.7), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=20
sage: for i in range(n):
... x = random()/2+.8; y = random()/2 - .25
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .02, 'r%s'%i)
...
...
sage: t.save()
r T # Worksheet 'bsd' (2006-08-27 at 05:30)
sage: t = Tachyon(xres=800, yres=600, camera_center=(1.5,.4,.7), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=100
sage: for i in range(n):
... x = random()/2+.8; y = random()/2 - .25
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .02, 'r%s'%i)
...
...
sage: t.save()
r! T # Worksheet 'bsd' (2006-08-27 at 05:30)
sage: t = Tachyon(xres=800, yres=600, camera_center=(1.5,.4,.7), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=100
sage: for i in range(n):
... x = random()/2+.8; y = random()/2 - .25
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .02, 'r%s'%i)
...
...
sage: t.save()
r" T # Worksheet 'bsd' (2006-08-27 at 05:31)
sage: t = Tachyon(xres=800, yres=600, camera_center=(1.5,.4,.7), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=200
sage: for i in range(n):
... x = random()/2+.8; y = random()/2 - .25
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .01, 'r%s'%i)
...
...
sage: t.save()
r# T> # Worksheet '_scratch_' (2006-08-27 at 05:32)
sage: E = EllipticCurve('37a')
sage: L = E.Lseries_dokchitser(5)
sage: t = Tachyon(xres=800, yres=600, camera_center=(1.5,.4,.7), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: def f(x,y):
... return abs(L(x+I*y))
r$ T # Worksheet 'bsd' (2006-08-27 at 05:32)
sage: t = Tachyon(xres=800, yres=600, camera_center=(1.5,.4,.7), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: n=500
sage: for i in range(n):
... x = random()/2+.8; y = random()/2 - .25
... z = L(x+I*y)
... m = abs(z)
... r = arg(z)
... t.texture('r%s'%i,color=(r/7,r,0))
... t.sphere((x,-y,m), .01, 'r%s'%i)
...
...
sage: t.save()
r% Tr # Worksheet 'bsd' (2006-08-27 at 05:34)
sage: L = E.Lseries_dokchitser(4)
sage: t = Tachyon(xres=800, yres=600, camera_center=(1.5,.4,.7), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: def f(x,y):
... return abs(L(x+I*y))
...
sage: t.plot(f, (0,1), (-.5,.5), 'red', max_depth=3, initial_depth=2)
sage: t.show()
r& Tr # Worksheet 'bsd' (2006-08-27 at 05:35)
sage: L = E.Lseries_dokchitser(4)
sage: t = Tachyon(xres=800, yres=600, camera_center=(1.5,.4,.7), look_at=(1,0,0), raydepth=4)
sage: t.light((10,3,2), 1, (1,1,1))
sage: t.light((10,-3,2), 1, (1,1,1))
sage: t.texture('black', color=(0,0,0))
sage: t.texture('red', color=(1,0,0))
sage: t.texture('grey', color=(.9,.9,.9))
sage: t.plane((0,0,0),(0,0,1),'grey')
sage: t.cylinder((0,0,0),(1,0,0),.01,'black')
sage: t.cylinder((0,0,0),(0,1,0),.01,'black')
sage: def f(x,y):
... return abs(L(x+I*y))
...
sage: t.plot(f, (0,1), (-.5,.5), 'red', max_depth=4, initial_depth=3)
sage: t.show()
r' To # Worksheet 'bsd' (2006-08-27 at 05:36)
hide%html> Conjecture (Birch and Swinnerton-Dyer): The order of vanishing of
hide%html> L(E,s) at s=1 equals the rank of E.
hide%html>
Conjecture (Birch and Swinnerton-Dyer): The order of vanishing of
L(E,s) at s=1 equals the rank of E.
r( T1 # Worksheet 'bsd' (2006-08-27 at 05:36)
hide%html> Conjecture (Birch and Swinnerton-Dyer): The order of vanishing of
hide%html> L(E,s) at s=1 equals the rank of E.
hide%html>
hide%html>
hide%html> This is known when the order of vanishing is \leq 1.
Conjecture (Birch and Swinnerton-Dyer): The order of vanishing of
L(E,s) at s=1 equals the rank of E.
This is known when the order of vanishing is \leq 1.r) TQ # Worksheet 'bsd' (2006-08-27 at 05:37)
hide%html> Conjecture (Birch and Swinnerton-Dyer): The order of vanishing of
hide%html> L(E,s) at s=1 equals the rank of E.
hide%html>
hide%html>
hide%html> This is known when the order of vanishing is \leq 1.
hide%html>
Conjecture (Birch and Swinnerton-Dyer): The order of vanishing of
L(E,s) at s=1 equals the rank of E.
This is known when the order of vanishing is \leq 1.
r* TY # Worksheet 'bsd' (2006-08-27 at 05:37)
hide%html> Conjecture (Birch and Swinnerton-Dyer): The order of vanishing of
hide%html> L(E,s) at s=1 equals the rank of E.
hide%html>
hide%html>
hide%html> This is a theorem when the order of vanishing is \leq 1.
hide%html>
Conjecture (Birch and Swinnerton-Dyer): The order of vanishing of
L(E,s) at s=1 equals the rank of E.
This is a theorem when the order of vanishing is \leq 1.
r+ TL # Worksheet 'bsd' (2006-08-27 at 05:38)
hide%html> Conjecture (Birch and Swinnerton-Dyer): The order of vanishing of
hide%html> L(E,s) at s=1 equals
hide%html> the rank of E.
hide%html>
hide%html>
hide%html> This is a theorem when the order of vanishing is \leq 1.
hide%html>
hide%html>
hide%html> There is also a conjecture about the first nonzero coefficient
hide%html> of the Taylor expansion of L(E,s)
hide%html> about s=1.
hide%html>
Conjecture (Birch and Swinnerton-Dyer): The order of vanishing of
L(E,s) at s=1 equals
the rank of E.
This is a theorem when the order of vanishing is \leq 1.
There is also a conjecture about the first nonzero coefficient
of the Taylor expansion of L(E,s)
about