#Rodney Keaton and Dania Zantout
|
|
<p>The following function is an implementation of a naive algorithm proposed by J. Achter. It is designed to calculate the rank of an elliptic curve of the form </p>
<p>$y^2=x^3+ax^2+bx.$</p>
The following function is an implementation of a naive algorithm proposed by J. Achter.
It is designed to calculate the rank of an elliptic curve of the form $y^2=x^3+ax^2+bx$.
def ECRank(a,b):
QQ(a)
QQ(b)
r = rnk(a,b)
rbar = rnkbar(a,b)
return r*rbar/2-1
def rnk(a,b):
x = QQ['x0,x1,x2'].gens()
candidates = b.squarefree_part().factor()
image = []
for b1 in candidates:
b2 = b/b1[0]
poly = b1[0]*x[0]^4+a*x[0]^2*x[1]^2+b2*x[1]^4-x[2]^2
for u in range(-100,100):
for v in range(-100,100):
for w in range(-100,100):
if poly(u,v,w) == 0:
image.append(b1[0])
break
if poly(u,v,w) == 0:
break
if poly(u,v,w) == 0:
break
return len(image)+1
def rnkbar(a,b):
x = QQ['x0,x1,x2'].gens()
abar = -2*a
bbar = a^2-4*b
candidates = bbar.squarefree_part().factor()
image = []
for b1 in candidates:
b2 = bbar/b1[0]
poly = b1[0]*x[0]^4+abar*x[0]^2*x[1]^2+b2*x[1]^4-x[2]^2
for u in range(-100,100):
for v in range(-100,100):
for w in range(-100,100):
if poly(u,v,w) == 0:
image.append(b1[0])
break
if poly(u,v,w) == 0:
break
if poly(u,v,w) == 0:
break
return len(image)+1
|
|
Let's run a test with the curve $y^2=x^3+x^2+x$.
ECRank(1,1)
0 0 |
timeit('ECRank(1,1)')
25 loops, best of 3: 14.3 ms per loop 25 loops, best of 3: 14.3 ms per loop |
E=EllipticCurve([0,1,0,1,0])
E.rank()
0 0 |
So, we obtained the correct rank in this case.
Now, let's try a curve with a higher rank, say $y^2=x^3+x^2+3x$.
timeit('E.rank()')
625 loops, best of 3: 4.45 µs per loop 625 loops, best of 3: 4.45 µs per loop |
So, we obtain the correct rank in this case.
Now, let's try a curve with a higher rank, say $y^2=x^3+x^2+2x$.
ECRank(1,2)
1 1 |
This took somewhere around 20 minutes to complete.
E=EllipticCurve([0,1,0,2,0])
E.rank()
1 1 |
At least we got the right answer.
timeit('E.rank()')
625 loops, best of 3: 4.47 µs per loop 625 loops, best of 3: 4.47 µs per loop |
Why is this so fast?
There are efficient ways of computing lower and upper bounds, and when they match, SAGE returns this as the rank.
print 'Upper Bound:',E.rank_bound()
print 'Lower Bound:',E.simon_two_descent()[0]
Upper Bound: 1 Lower Bound: 1 Upper Bound: 1 Lower Bound: 1 |
Now, lets compare some of the algorithms that SAGE has built in to compute the rank.
RANK 0
E=EllipticCurve([0,1,0,1,0])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 1001568 (0.955 Mbytes). *** Warning: new stack size = 1001568 (0.955 Mbytes). *** Warning: new stack size = 1001568 (0.955 Mbytes). 1 loops, best of 3: 18.7 ms per loop *** Warning: new stack size = 1001568 (0.955 Mbytes). *** Warning: new stack size = 1001568 (0.955 Mbytes). *** Warning: new stack size = 1001568 (0.955 Mbytes). 1 loops, best of 3: 18.7 ms per loop |
timeit("E.analytic_rank(algorithm='sympow')",number=1)
1 loops, best of 3: 34.2 ms per loop 1 loops, best of 3: 34.2 ms per loop |
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 274 µs per loop 1 loops, best of 3: 274 µs per loop |
timeit('E.rank()')
625 loops, best of 3: 4.15 µs per loop 625 loops, best of 3: 4.15 µs per loop |
RANK 1
E=EllipticCurve([0,1,0,2,0])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 1002672 (0.956 Mbytes). *** Warning: new stack size = 1002672 (0.956 Mbytes). *** Warning: new stack size = 1002672 (0.956 Mbytes). 1 loops, best of 3: 19.5 ms per loop *** Warning: new stack size = 1002672 (0.956 Mbytes). *** Warning: new stack size = 1002672 (0.956 Mbytes). *** Warning: new stack size = 1002672 (0.956 Mbytes). 1 loops, best of 3: 19.5 ms per loop |
timeit("E.analytic_rank(algorithm='sympow')",number=1)
1 loops, best of 3: 34.2 ms per loop 1 loops, best of 3: 34.2 ms per loop |
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 2.59 ms per loop 1 loops, best of 3: 2.59 ms per loop |
timeit('E.rank()')
625 loops, best of 3: 4.17 µs per loop 625 loops, best of 3: 4.17 µs per loop |
RANK 2
E=EllipticCurve([0,1,1,-2,0])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 1003360 (0.957 Mbytes). *** Warning: new stack size = 1003360 (0.957 Mbytes). *** Warning: new stack size = 1003360 (0.957 Mbytes). 1 loops, best of 3: 21.3 ms per loop *** Warning: new stack size = 1003360 (0.957 Mbytes). *** Warning: new stack size = 1003360 (0.957 Mbytes). *** Warning: new stack size = 1003360 (0.957 Mbytes). 1 loops, best of 3: 21.3 ms per loop |
timeit("E.analytic_rank(algorithm='sympow')",number=1)
1 loops, best of 3: 42.3 ms per loop 1 loops, best of 3: 42.3 ms per loop |
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 6.5 ms per loop 1 loops, best of 3: 6.5 ms per loop |
timeit("E.rank()",number=1)
1 loops, best of 3: 9.06 µs per loop 1 loops, best of 3: 9.06 µs per loop |
RANK 3
E=EllipticCurve([0,0,1,-7,6])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 1010496 (0.964 Mbytes). *** Warning: new stack size = 1010496 (0.964 Mbytes). *** Warning: new stack size = 1010496 (0.964 Mbytes). 1 loops, best of 3: 24.3 ms per loop *** Warning: new stack size = 1010496 (0.964 Mbytes). *** Warning: new stack size = 1010496 (0.964 Mbytes). *** Warning: new stack size = 1010496 (0.964 Mbytes). 1 loops, best of 3: 24.3 ms per loop |
timeit("E.analytic_rank(algorithm='sympow')",number=1)
1 loops, best of 3: 40.5 ms per loop 1 loops, best of 3: 40.5 ms per loop |
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 23.6 ms per loop 1 loops, best of 3: 23.6 ms per loop |
timeit("E.rank()",number=1)
1 loops, best of 3: 9.06 µs per loop 1 loops, best of 3: 9.06 µs per loop |
RANK 4
E=EllipticCurve([1,-1,0,-79,289])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 1067712 (1.018 Mbytes). *** Warning: new stack size = 1067712 (1.018 Mbytes). *** Warning: new stack size = 1067712 (1.018 Mbytes). 1 loops, best of 3: 57.3 ms per loop *** Warning: new stack size = 1067712 (1.018 Mbytes). *** Warning: new stack size = 1067712 (1.018 Mbytes). *** Warning: new stack size = 1067712 (1.018 Mbytes). 1 loops, best of 3: 57.3 ms per loop |
timeit("E.analytic_rank(algorithm='sympow')",number=1)
1 loops, best of 3: 47.1 ms per loop 1 loops, best of 3: 47.1 ms per loop |
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 171 ms per loop 1 loops, best of 3: 171 ms per loop |
timeit("E.rank()",number=1)
1 loops, best of 3: 7.87 µs per loop 1 loops, best of 3: 7.87 µs per loop |
RANK 5
E=EllipticCurve([0,0,1,-79,342])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 1786816 (1.704 Mbytes). *** Warning: new stack size = 1786816 (1.704 Mbytes). *** Warning: new stack size = 1786816 (1.704 Mbytes). 1 loops, best of 3: 561 ms per loop *** Warning: new stack size = 1786816 (1.704 Mbytes). *** Warning: new stack size = 1786816 (1.704 Mbytes). *** Warning: new stack size = 1786816 (1.704 Mbytes). 1 loops, best of 3: 561 ms per loop |
timeit("E.analytic_rank(algorithm='sympow')",number=1)
/usr/local/sage/local/bin/sympow: line 3: 46927 Segmentation fault (core dumped) ./sympow $* Traceback (click to the left of this block for traceback) ... RuntimeError: failed to compute analytic rank /usr/local/sage/local/bin/sympow: line 3: 46927 Segmentation fault (core dumped) ./sympow $*
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_60.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("dGltZWl0KCJFLmFuYWx5dGljX3JhbmsoYWxnb3JpdGhtPSdzeW1wb3cnKSIsbnVtYmVyPTEp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpXrbeP9/___code___.py", line 3, in <module>
exec compile(u'timeit("E.analytic_rank(algorithm=\'sympow\')",number=_sage_const_1 )
File "", line 1, in <module>
File "sage_timeit_class.pyx", line 82, in sage.misc.sage_timeit_class.SageTimeit.__call__ (sage/misc/sage_timeit_class.c:744)
File "sage_timeit_class.pyx", line 59, in sage.misc.sage_timeit_class.SageTimeit.eval (sage/misc/sage_timeit_class.c:605)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/misc/sage_timeit.py", line 184, in sage_timeit
best = min(timer.repeat(repeat, number)) / number
File "/usr/local/sage/local/lib/python/timeit.py", line 220, in repeat
t = self.timeit(number)
File "/usr/local/sage/local/lib/python/timeit.py", line 193, in timeit
timing = self.inner(it, self.timer)
File "<magic-timeit>", line 6, in inner
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_rational_field.py", line 1373, in analytic_rank
return sympow.analytic_rank(self)[0]
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/lfunctions/sympow.py", line 287, in analytic_rank
raise RuntimeError, "failed to compute analytic rank"
RuntimeError: failed to compute analytic rank
|
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 1.77 s per loop 1 loops, best of 3: 1.77 s per loop |
timeit("E.rank()",number=1)
1 loops, best of 3: 9.06 µs per loop 1 loops, best of 3: 9.06 µs per loop |
RANK 6
E=EllipticCurve([1,1,0,-2582,48720])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 13975616 (13.328 Mbytes). *** Warning: new stack size = 13975616 (13.328 Mbytes). *** Warning: new stack size = 13975616 (13.328 Mbytes). 1 loops, best of 3: 9.03 s per loop *** Warning: new stack size = 13975616 (13.328 Mbytes). *** Warning: new stack size = 13975616 (13.328 Mbytes). *** Warning: new stack size = 13975616 (13.328 Mbytes). 1 loops, best of 3: 9.03 s per loop |
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 35.1 s per loop 1 loops, best of 3: 35.1 s per loop |
timeit("E.rank()",number=1)
1 loops, best of 3: 8.82 µs per loop 1 loops, best of 3: 8.82 µs per loop |
RANK 7
E=EllipticCurve([0,0,0,-10012,346900])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 112433152 (107.225 Mbytes). *** Warning: new stack size = 112433152 (107.225 Mbytes). *** Warning: new stack size = 112433152 (107.225 Mbytes). 1 loops, best of 3: 57.2 s per loop *** Warning: new stack size = 112433152 (107.225 Mbytes). *** Warning: new stack size = 112433152 (107.225 Mbytes). *** Warning: new stack size = 112433152 (107.225 Mbytes). 1 loops, best of 3: 57.2 s per loop |
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 186 s per loop 1 loops, best of 3: 186 s per loop |
timeit("E.rank()",number=1)
1 loops, best of 3: 11 µs per loop 1 loops, best of 3: 11 µs per loop |
RANK 8
E=EllipticCurve([0,0,1,-23737,960366])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: not enough memory, new stack 18446744073268923520. *** Warning: not enough memory, new stack 9223372036634461760. *** Warning: not enough memory, new stack 4611686018317230880. *** Warning: not enough memory, new stack 2305843009158615440. *** Warning: not enough memory, new stack 1152921504579307720. *** Warning: not enough memory, new stack 576460752289653860. *** Warning: not enough memory, new stack 288230376144826930. *** Warning: not enough memory, new stack 144115188072413465. *** Warning: not enough memory, new stack 72057594036206732. *** Warning: not enough memory, new stack 36028797018103366. *** Warning: not enough memory, new stack 18014398509051683. *** Warning: not enough memory, new stack 9007199254525841. *** Warning: not enough memory, new stack 4503599627262920. *** Warning: not enough memory, new stack 2251799813631460. *** Warning: not enough memory, new stack 1125899906815730. *** Warning: not enough memory, new stack 562949953407865. *** Warning: not enough memory, new stack 281474976703932. *** Warning: not enough memory, new stack 140737488351966. *** Warning: not enough memory, new stack 70368744175983. *** Warning: not enough memory, new stack 35184372087991. *** Warning: not enough memory, new stack 17592186043995. *** Warning: not enough memory, new stack 8796093021997. *** Warning: not enough memory, new stack 4398046510998. *** Warning: not enough memory, new stack 2199023255499. *** Warning: not enough memory, new stack 1099511627749. *** Warning: not enough memory, new stack 549755813874. *** Warning: not enough memory, new stack 274877906937. *** Warning: not enough memory, new stack 137438953468. *** Warning: not enough memory, new stack 68719476734. *** Warning: new stack size = 34359738367 (32768.000 Mbytes). *** bug in PARI/GP (Segmentation Fault), please report Traceback (click to the left of this block for traceback) ... __SAGE__ *** Warning: not enough memory, new stack 18446744073268923520.
*** Warning: not enough memory, new stack 9223372036634461760.
*** Warning: not enough memory, new stack 4611686018317230880.
*** Warning: not enough memory, new stack 2305843009158615440.
*** Warning: not enough memory, new stack 1152921504579307720.
*** Warning: not enough memory, new stack 576460752289653860.
*** Warning: not enough memory, new stack 288230376144826930.
*** Warning: not enough memory, new stack 144115188072413465.
*** Warning: not enough memory, new stack 72057594036206732.
*** Warning: not enough memory, new stack 36028797018103366.
*** Warning: not enough memory, new stack 18014398509051683.
*** Warning: not enough memory, new stack 9007199254525841.
*** Warning: not enough memory, new stack 4503599627262920.
*** Warning: not enough memory, new stack 2251799813631460.
*** Warning: not enough memory, new stack 1125899906815730.
*** Warning: not enough memory, new stack 562949953407865.
*** Warning: not enough memory, new stack 281474976703932.
*** Warning: not enough memory, new stack 140737488351966.
*** Warning: not enough memory, new stack 70368744175983.
*** Warning: not enough memory, new stack 35184372087991.
*** Warning: not enough memory, new stack 17592186043995.
*** Warning: not enough memory, new stack 8796093021997.
*** Warning: not enough memory, new stack 4398046510998.
*** Warning: not enough memory, new stack 2199023255499.
*** Warning: not enough memory, new stack 1099511627749.
*** Warning: not enough memory, new stack 549755813874.
*** Warning: not enough memory, new stack 274877906937.
*** Warning: not enough memory, new stack 137438953468.
*** Warning: not enough memory, new stack 68719476734.
*** Warning: new stack size = 34359738367 (32768.000 Mbytes).
*** bug in PARI/GP (Segmentation Fault), please report
^CTraceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_99.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("dGltZWl0KCJFLmFuYWx5dGljX3JhbmsoYWxnb3JpdGhtPSdydWJpbnN0ZWluJykiLG51bWJlcj0xKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpGVvD8B/___code___.py", line 3, in <module>
exec compile(u'timeit("E.analytic_rank(algorithm=\'rubinstein\')",number=_sage_const_1 )
File "", line 1, in <module>
File "sage_timeit_class.pyx", line 82, in sage.misc.sage_timeit_class.SageTimeit.__call__ (sage/misc/sage_timeit_class.c:744)
File "sage_timeit_class.pyx", line 59, in sage.misc.sage_timeit_class.SageTimeit.eval (sage/misc/sage_timeit_class.c:605)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/misc/sage_timeit.py", line 184, in sage_timeit
best = min(timer.repeat(repeat, number)) / number
File "/usr/local/sage/local/lib/python/timeit.py", line 220, in repeat
t = self.timeit(number)
File "/usr/local/sage/local/lib/python/timeit.py", line 193, in timeit
timing = self.inner(it, self.timer)
File "<magic-timeit>", line 6, in inner
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_rational_field.py", line 1366, in analytic_rank
return lcalc.analytic_rank(L=self)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/lfunctions/lcalc.py", line 379, in analytic_rank
s = self('--rank-compute %s'%L)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/lfunctions/lcalc.py", line 65, in __call__
return os.popen(cmd).read().strip()
KeyboardInterrupt
__SAGE__
|
timeit("E.rank()",number=1)
1 loops, best of 3: 10 µs per loop 1 loops, best of 3: 10 µs per loop |
#Rubinstein in Red
#Sympow in Blue
#Pari in Black
#Rank bound method in Green
data1 = [(0,0.0187),(1,0.0195), (2,0.0213), (3,0.0243), (4,0.0598)]
g1 = Graphics()
g1 += point2d(data1,color='red')
g1 += line(data1,color='red')
data2 = [(0,0.0342),(1,0.0342), (2,0.0423), (3,0.0405), (4,0.0471)]
g2 = Graphics()
g2 += point2d(data2,color='blue')
g2 += line(data2,color='blue')
data3 = [(0,0.000274),(1,0.00259), (2,0.0065), (3,0.0236), (4,0.171)]
g3 = Graphics()
g3 += point2d(data3,color='black')
g3 += line(data3,color='black')
show(g1+g2+g3)
|
|
data1 = [(0,0.0187),(1,0.0195), (2,0.0213), (3,0.0243), (4,0.0598),(5,0.561), (6,9.03), (7,57.2)]
g1 = Graphics()
g1 += point2d(data1,color='red')
g1 += line(data1,color='red')
data2 = [(0,0.00000415),(1,0.00000417), (2,0.00000906), (3,0.00000906), (4,0.00000787),(5,0.00000906),(6,0.00000882),(7,0.000011)]
g2 = Graphics()
g2 += point2d(data2,color='green')
g2 += line(data2,color='green')
data3 = [(0,0.000274),(1,0.00259), (2,0.0065), (3,0.0236), (4,0.171),(5,1.77),(6,35.1),(7,186)]
g3 = Graphics()
g3 += point2d(data3,color='black')
g3 += line(data3,color='black')
show(g1+g2+g3)
|
|
What about some cases when the upper and lower bound do not match?
RANK 0
E=EllipticCurve([-1,-1,-1,3,7])
E.rank()
Unable to compute the rank with certainty (lower bound=0). This could be because Sha(E/Q)[2] is nontrivial. Try calling something like two_descent(second_limit=13) on the curve then trying this command again. You could also try rank with only_use_mwrank=False. Traceback (click to the left of this block for traceback) ... RuntimeError: Rank not provably correct. Unable to compute the rank with certainty (lower bound=0).
This could be because Sha(E/Q)[2] is nontrivial.
Try calling something like two_descent(second_limit=13) on the
curve then trying this command again. You could also try rank
with only_use_mwrank=False.
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_43.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("RT1FbGxpcHRpY0N1cnZlKFstMSwtMSwtMSwzLDddKQpFLnJhbmsoKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmp75FKMW/___code___.py", line 4, in <module>
exec compile(u'E.rank()
File "", line 1, in <module>
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_rational_field.py", line 1695, in rank
raise RuntimeError, 'Rank not provably correct.'
RuntimeError: Rank not provably correct.
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 1024848 (0.977 Mbytes). *** Warning: new stack size = 1024848 (0.977 Mbytes). *** Warning: new stack size = 1024848 (0.977 Mbytes). 1 loops, best of 3: 24.7 ms per loop *** Warning: new stack size = 1024848 (0.977 Mbytes). *** Warning: new stack size = 1024848 (0.977 Mbytes). *** Warning: new stack size = 1024848 (0.977 Mbytes). 1 loops, best of 3: 24.7 ms per loop |
timeit("E.analytic_rank(algorithm='sympow')",number=1)
1 loops, best of 3: 31.4 ms per loop 1 loops, best of 3: 31.4 ms per loop |
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 6.74 ms per loop 1 loops, best of 3: 6.74 ms per loop |
RANK 1
E=EllipticCurve([-1,1,0,8,9])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 1028800 (0.981 Mbytes). *** Warning: new stack size = 1028800 (0.981 Mbytes). *** Warning: new stack size = 1028800 (0.981 Mbytes). 1 loops, best of 3: 29.2 ms per loop *** Warning: new stack size = 1028800 (0.981 Mbytes). *** Warning: new stack size = 1028800 (0.981 Mbytes). *** Warning: new stack size = 1028800 (0.981 Mbytes). 1 loops, best of 3: 29.2 ms per loop |
timeit("E.analytic_rank(algorithm='sympow')",number=1)
1 loops, best of 3: 31.8 ms per loop 1 loops, best of 3: 31.8 ms per loop |
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 32.2 ms per loop 1 loops, best of 3: 32.2 ms per loop |
RANK 3
E=EllipticCurve([0,157,0,-160,1])
|
|
timeit("E.analytic_rank(algorithm='rubinstein')",number=1)
*** Warning: new stack size = 13804640 (13.165 Mbytes). *** Warning: new stack size = 13804640 (13.165 Mbytes). *** Warning: new stack size = 13804640 (13.165 Mbytes). 1 loops, best of 3: 4.66 s per loop *** Warning: new stack size = 13804640 (13.165 Mbytes). *** Warning: new stack size = 13804640 (13.165 Mbytes). *** Warning: new stack size = 13804640 (13.165 Mbytes). 1 loops, best of 3: 4.66 s per loop |
timeit("E.analytic_rank(algorithm='sympow')",number=1)
1 loops, best of 3: 291 ms per loop 1 loops, best of 3: 291 ms per loop |
timeit("E.analytic_rank(algorithm='pari')",number=1)
1 loops, best of 3: 14.3 s per loop 1 loops, best of 3: 14.3 s per loop |
data1 = [(0,0.0217),(1,0.0292), (3,4.66)]
g1 = Graphics()
g1 += point2d(data1,color='red')
g1 += line(data1,color='red')
data2 = [(0,0.0314),(1,0.0318), (3,0.291)]
g2 = Graphics()
g2 += point2d(data2,color='blue')
g2 += line(data2,color='blue')
data3 = [(0,0.00674),(1,0.0322), (3,14.3)]
g3 = Graphics()
g3 += point2d(data3,color='black')
g3 += line(data3,color='black')
show(g1+g2+g3)
|
|
|
|