N = 1000
E = EllipticCurve([0, 0, 0, -1, 0])
champion = []
rest = []
for p in prime_range(N):
if E.has_good_reduction(p):
trace = E.reduction(p).trace_of_frobenius()
if trace <= -floor(2 * sqrt(p)):
champion.append((p, 1 + p - trace))
else:
rest.append((p, 1 + p - trace))
championPoints = point2d(champion, rgbcolor = (1, 0, 0))
restPoints = point2d(rest)
show(championPoints + restPoints + plot(1 + x + floor(2 * sqrt(x)), (x, 0, N)))
|
E.has_cm()
True True |
def normalizedTracePlot(E, N):
all = []
for p in prime_range(N):
if E.has_good_reduction(p):
trace = E.reduction(p).trace_of_frobenius()
all.append((p, trace / floor(2 * sqrt(p))))
allPoints = point2d(all)
show(allPoints)
return
|
|
N = 5000
normalizedTracePlot(EllipticCurve([0,0,0,0,17]), N)
|
normalizedTracePlot(EllipticCurve([0,0,0,-1,0]), N)
|
normalizedTracePlot(EllipticCurve([0,0,0,1,3]), N)
|
N = 300
spacing = 10
loA = -10
hiA = 10
loB = -10
hiB = 10
for a in range(loA, hiA):
for b in range(loB, hiB):
if (4*a*a*a + 27*b*b != 0): # not singular
E = EllipticCurve([0,0,0,a,b])
s = '(' + repr(a) + ', ' + repr(b) + ')'
if E.has_cm():
cm = 'CM'
else:
cm = ''
print s.ljust(spacing), cm.ljust(spacing),
flag = ''
for p in prime_range(N):
if E.has_good_reduction(p):
if E.reduction(p).trace_of_frobenius() == -floor(2 * sqrt(p)):
flag += repr(p).ljust(spacing)
# break
print flag
WARNING: Output truncated!full_output.txt (-10, -10) (-10, -9) (-10, -8) 3 11 67 271 (-10, -7) (-10, -6) (-10, -5) 3 (-10, -4) (-10, -3) 19 (-10, -2) 3 (-10, -1) 53 (-10, 0) CM 101 109 173 (-10, 1) 3 53 (-10, 2) (-10, 3) 11 (-10, 4) 3 (-10, 5) 127 (-10, 6) 179 (-10, 7) 3 23 (-10, 8) (-10, 9) (-9, -10) 41 (-9, -9) (-9, -8) 107 271 (-9, -7) (-9, -6) (-9, -5) (-9, -4) (-9, -3) (-9, -2) (-9, -1) (-9, 0) CM 53 (-9, 1) (-9, 2) (-9, 3) (-9, 4) (-9, 5) 131 (-9, 6) (-9, 7) (-9, 8) (-9, 9) (-8, -10) 11 (-8, -9) (-8, -8) (-8, -7) (-8, -6) (-8, -5) (-8, -4) (-8, -3) 103 (-8, -2) 19 (-8, -1) 41 (-8, 0) CM 37 197 (-8, 1) 11 41 (-8, 2) (-8, 3) (-8, 4) (-8, 5) (-8, 6) (-8, 7) (-8, 8) ... (7, -10) (7, -9) (7, -8) (7, -7) 53 107 (7, -6) (7, -5) (7, -4) 7 (7, -3) (7, -2) 19 (7, -1) (7, 0) CM 257 (7, 1) (7, 2) (7, 3) 7 (7, 4) (7, 5) (7, 6) (7, 7) 53 (7, 8) (7, 9) (8, -10) 5 (8, -9) 41 (8, -8) 3 (8, -7) (8, -6) 23 (8, -5) 3 5 79 (8, -4) 89 (8, -3) 271 (8, -2) 3 29 (8, -1) (8, 0) CM 5 101 109 (8, 1) 3 59 67 (8, 2) 29 (8, 3) (8, 4) 3 89 (8, 5) 5 139 (8, 6) (8, 7) 3 (8, 8) (8, 9) 41 (9, -10) (9, -9) (9, -8) 41 47 (9, -7) 11 (9, -6) 53 (9, -5) (9, -4) 79 179 (9, -3) 19 (9, -2) 23 127 (9, -1) (9, 0) CM 29 173 229 293 (9, 1) 151 (9, 2) (9, 3) (9, 4) 11 (9, 5) (9, 6) 53 (9, 7) (9, 8) 41 (9, 9) WARNING: Output truncated!full_output.txt (-10, -10) (-10, -9) (-10, -8) 3 11 67 271 (-10, -7) (-10, -6) (-10, -5) 3 (-10, -4) (-10, -3) 19 (-10, -2) 3 (-10, -1) 53 (-10, 0) CM 101 109 173 (-10, 1) 3 53 (-10, 2) (-10, 3) 11 (-10, 4) 3 (-10, 5) 127 (-10, 6) 179 (-10, 7) 3 23 (-10, 8) (-10, 9) (-9, -10) 41 (-9, -9) (-9, -8) 107 271 (-9, -7) (-9, -6) (-9, -5) (-9, -4) (-9, -3) (-9, -2) (-9, -1) (-9, 0) CM 53 (-9, 1) (-9, 2) (-9, 3) (-9, 4) (-9, 5) 131 (-9, 6) (-9, 7) (-9, 8) (-9, 9) (-8, -10) 11 (-8, -9) (-8, -8) (-8, -7) (-8, -6) (-8, -5) (-8, -4) (-8, -3) 103 (-8, -2) 19 (-8, -1) 41 (-8, 0) CM 37 197 (-8, 1) 11 41 (-8, 2) (-8, 3) (-8, 4) (-8, 5) (-8, 6) (-8, 7) (-8, 8) ... (7, -10) (7, -9) (7, -8) (7, -7) 53 107 (7, -6) (7, -5) (7, -4) 7 (7, -3) (7, -2) 19 (7, -1) (7, 0) CM 257 (7, 1) (7, 2) (7, 3) 7 (7, 4) (7, 5) (7, 6) (7, 7) 53 (7, 8) (7, 9) (8, -10) 5 (8, -9) 41 (8, -8) 3 (8, -7) (8, -6) 23 (8, -5) 3 5 79 (8, -4) 89 (8, -3) 271 (8, -2) 3 29 (8, -1) (8, 0) CM 5 101 109 (8, 1) 3 59 67 (8, 2) 29 (8, 3) (8, 4) 3 89 (8, 5) 5 139 (8, 6) (8, 7) 3 (8, 8) (8, 9) 41 (9, -10) (9, -9) (9, -8) 41 47 (9, -7) 11 (9, -6) 53 (9, -5) (9, -4) 79 179 (9, -3) 19 (9, -2) 23 127 (9, -1) (9, 0) CM 29 173 229 293 (9, 1) 151 (9, 2) (9, 3) (9, 4) 11 (9, 5) (9, 6) 53 (9, 7) (9, 8) 41 (9, 9) |
N = 1000
M = 20
spacing = 11
for n in range(-M, M):
if (n != 0 and n != 1): # not singular
E = EllipticCurve([0, -(n + 1), 0, n, 0]) # Legendre form
j = round(E.j_invariant().numerical_approx(), ndigits = 2)
if E.has_cm():
cm = 'CM'
else:
cm = ''
print repr(n).ljust(spacing), repr(j).ljust(spacing), cm.ljust(spacing),
flag = ''
for p in prime_range(N):
if E.has_good_reduction(p):
if E.reduction(p).trace_of_frobenius() == -floor(2 * sqrt(p)):
if E.is_supersingular(p):
flag += 'SS'
flag += repr(p).ljust(spacing)
print flag
-20 108289.83 -19 98050.02 -18 88322.25 -17 79106.51 -16 70402.83 -15 62211.2 -14 54531.66 -13 47364.23 -12 40708.93 89 -11 34565.83 19 -10 28935.0 -9 23816.56 419 -8 19210.72 -7 15117.8 -6 11538.43 -5 8473.88 -4 5927.04 -3 3905.78 -2 2439.11 71 691 -1 1728.0 CM 29 173 293 977 2 1728.0 CM 29 173 293 977 3 2439.11 4 3905.78 5 5927.04 587 6 8473.88 7 11538.43 271 8 15117.8 19 9 19210.72 10 23816.56 11 28935.0 107 12 34565.83 13 40708.93 89 331 14 47364.23 151 15 54531.66 503 16 62211.2 17 70402.83 18 79106.51 19 88322.25 -20 108289.83 -19 98050.02 -18 88322.25 -17 79106.51 -16 70402.83 -15 62211.2 -14 54531.66 -13 47364.23 -12 40708.93 89 -11 34565.83 19 -10 28935.0 -9 23816.56 419 -8 19210.72 -7 15117.8 -6 11538.43 -5 8473.88 -4 5927.04 -3 3905.78 -2 2439.11 71 691 -1 1728.0 CM 29 173 293 977 2 1728.0 CM 29 173 293 977 3 2439.11 4 3905.78 5 5927.04 587 6 8473.88 7 11538.43 271 8 15117.8 19 9 19210.72 10 23816.56 11 28935.0 107 12 34565.83 13 40708.93 89 331 14 47364.23 151 15 54531.66 503 16 62211.2 17 70402.83 18 79106.51 19 88322.25 |
N = 10000
spacing = 8
E = EllipticCurve([0, 0, 0, 7, 9])
for p in prime_range(N):
if E.has_good_reduction(p):
if E.reduction(p).trace_of_frobenius() == -floor(2 * sqrt(p)):
print repr(p)
2069 2069 |
def printChampion(E, N):
list = []
for p in prime_range(N):
if E.has_good_reduction(p):
if E.reduction(p).trace_of_frobenius() == -floor(2 * sqrt(p)):
list.append(p)
return list
|
|
N = 1000
M = 50
for b in range(1, M):
E = EllipticCurve([0, 0, 0, 0, b])
print repr(b).ljust(4), printChampion(E, N)
1 [67, 199, 541, 853] 2 [97, 139, 607, 709, 811, 877, 937] 3 [7, 31, 79, 97, 103, 157, 241, 571, 787, 853] 4 [13, 241, 463, 619, 757] 5 [73, 199, 307, 487, 877] 6 [31, 181] 7 [181, 487, 541, 619] 8 [19, 181, 199, 373, 787, 811] 9 [13, 43, 67, 163, 349, 463, 601, 607, 853, 937] 10 [7, 103, 157, 163, 313, 571, 757] 11 [73, 157, 607, 757] 12 [19, 31, 139, 157, 349, 487, 757, 937] 13 [43, 307, 757] 14 [43, 67, 313, 607, 811] 15 [43, 67, 73, 139, 241, 349, 439, 541, 601, 607, 811] 16 [97, 163, 349, 607, 937] 17 [7, 13, 31, 43, 157, 313, 439, 463, 709, 757, 937] 18 [19, 199, 241, 373, 421, 571, 601] 19 [139, 157, 181, 373, 571, 607, 757] 20 [241, 307, 349, 373, 541, 787] 21 [139] 22 [13, 67, 103, 181, 307, 349, 421, 619, 853] 23 [349, 373, 709] 24 [7, 31, 67, 79, 97, 103, 241, 421, 709] 25 [43, 67, 199, 349, 601, 607, 619, 757, 853] 26 [73, 163, 181, 487] 27 [19, 103, 487, 541, 787, 811, 853] 28 [73, 79, 199, 571] 29 [211, 463] 30 [13, 79, 181, 307, 421, 439, 601, 619, 757] 31 [7, 19, 97, 103, 181, 313, 439, 709, 937] 32 [211, 241, 421, 463, 571, 811] 33 [73, 163, 307, 373, 463, 757] 34 [31, 79, 163, 421, 439, 487, 853] 35 [13, 79, 157, 163, 181, 463, 853] 36 [43, 97, 541, 601, 619, 757] 37 [19, 31, 79, 103, 313, 439, 487, 601, 619, 709] 38 [7, 211, 787, 811] 39 [103, 421, 463, 709, 757, 877, 937] 40 [67, 73, 157, 181, 199, 421, 487, 709, 853] 41 [163, 211, 439, 853] 42 [103, 157, 349, 463, 877] 43 [13, 31, 79, 97, 139, 307, 313, 463, 487, 607, 787] 44 [97, 157, 487] 45 [7, 73, 307, 313, 373] 46 [19, 163, 313, 787] 47 [73, 163, 313, 571, 811] 48 [13, 31, 157, 211, 541] 49 [241, 541, 601, 757, 853, 937] 1 [67, 199, 541, 853] 2 [97, 139, 607, 709, 811, 877, 937] 3 [7, 31, 79, 97, 103, 157, 241, 571, 787, 853] 4 [13, 241, 463, 619, 757] 5 [73, 199, 307, 487, 877] 6 [31, 181] 7 [181, 487, 541, 619] 8 [19, 181, 199, 373, 787, 811] 9 [13, 43, 67, 163, 349, 463, 601, 607, 853, 937] 10 [7, 103, 157, 163, 313, 571, 757] 11 [73, 157, 607, 757] 12 [19, 31, 139, 157, 349, 487, 757, 937] 13 [43, 307, 757] 14 [43, 67, 313, 607, 811] 15 [43, 67, 73, 139, 241, 349, 439, 541, 601, 607, 811] 16 [97, 163, 349, 607, 937] 17 [7, 13, 31, 43, 157, 313, 439, 463, 709, 757, 937] 18 [19, 199, 241, 373, 421, 571, 601] 19 [139, 157, 181, 373, 571, 607, 757] 20 [241, 307, 349, 373, 541, 787] 21 [139] 22 [13, 67, 103, 181, 307, 349, 421, 619, 853] 23 [349, 373, 709] 24 [7, 31, 67, 79, 97, 103, 241, 421, 709] 25 [43, 67, 199, 349, 601, 607, 619, 757, 853] 26 [73, 163, 181, 487] 27 [19, 103, 487, 541, 787, 811, 853] 28 [73, 79, 199, 571] 29 [211, 463] 30 [13, 79, 181, 307, 421, 439, 601, 619, 757] 31 [7, 19, 97, 103, 181, 313, 439, 709, 937] 32 [211, 241, 421, 463, 571, 811] 33 [73, 163, 307, 373, 463, 757] 34 [31, 79, 163, 421, 439, 487, 853] 35 [13, 79, 157, 163, 181, 463, 853] 36 [43, 97, 541, 601, 619, 757] 37 [19, 31, 79, 103, 313, 439, 487, 601, 619, 709] 38 [7, 211, 787, 811] 39 [103, 421, 463, 709, 757, 877, 937] 40 [67, 73, 157, 181, 199, 421, 487, 709, 853] 41 [163, 211, 439, 853] 42 [103, 157, 349, 463, 877] 43 [13, 31, 79, 97, 139, 307, 313, 463, 487, 607, 787] 44 [97, 157, 487] 45 [7, 73, 307, 313, 373] 46 [19, 163, 313, 787] 47 [73, 163, 313, 571, 811] 48 [13, 31, 157, 211, 541] 49 [241, 541, 601, 757, 853, 937] |
N = 1000
M = 20
for a in range(-M, M):
if a != 0:
E = EllipticCurve([0, 0, 0, a, 0])
print repr(a).ljust(4), printChampion(E, N)
-20 [17, 29, 197, 577, 641] -19 [37, 257, 401, 409, 733] -18 [109, 197, 457, 977] -17 [5, 37, 53, 109, 197, 293, 401, 733, 809] -16 [29, 173, 293, 977] -15 [101, 229, 577, 809, 857] -14 [17, 109, 173, 197, 257, 409, 457, 577, 701, 733, 977] -13 [37, 401, 409, 857] -12 [5, 17, 197, 229, 257, 457, 677, 733] -11 [53, 101, 229, 409, 809] -10 [101, 109, 173, 577] -9 [53, 641, 857] -8 [37, 197, 457, 641, 857] -7 [5, 29, 53, 101, 257, 577, 641] -6 [37, 53, 109, 173, 257, 401, 809] -5 [17, 37, 229, 577, 641, 677] -4 [53, 229, 733, 977] -3 [17, 101, 257, 457, 701] -2 [5, 101, 109, 457, 641, 677, 701, 857] -1 [29, 173, 293, 977] 1 [53, 229, 733, 977] 2 [37, 197, 457, 641, 857] 3 [5, 17, 197, 229, 257, 457, 677, 733] 4 [29, 173, 293, 977] 5 [17, 29, 197, 577, 641] 6 [29, 257, 293, 401, 809] 7 [257, 577, 641, 677] 8 [5, 101, 109, 457, 641, 677, 701, 857] 9 [29, 173, 229, 293, 641, 733, 857] 10 [53, 293, 577, 701, 733] 11 [109, 197, 409, 677, 701, 809] 12 [17, 101, 257, 457, 701] 13 [5, 29, 53, 109, 173, 197, 401, 409, 857] 14 [17, 37, 229, 257, 293, 409, 457, 577, 977] 15 [37, 53, 173, 293, 577, 701, 809, 857] 16 [53, 229, 733, 977] 17 [229, 401, 677, 809] 18 [5, 37, 101, 457, 677, 701, 977] 19 [109, 229, 257, 401, 409, 677] -20 [17, 29, 197, 577, 641] -19 [37, 257, 401, 409, 733] -18 [109, 197, 457, 977] -17 [5, 37, 53, 109, 197, 293, 401, 733, 809] -16 [29, 173, 293, 977] -15 [101, 229, 577, 809, 857] -14 [17, 109, 173, 197, 257, 409, 457, 577, 701, 733, 977] -13 [37, 401, 409, 857] -12 [5, 17, 197, 229, 257, 457, 677, 733] -11 [53, 101, 229, 409, 809] -10 [101, 109, 173, 577] -9 [53, 641, 857] -8 [37, 197, 457, 641, 857] -7 [5, 29, 53, 101, 257, 577, 641] -6 [37, 53, 109, 173, 257, 401, 809] -5 [17, 37, 229, 577, 641, 677] -4 [53, 229, 733, 977] -3 [17, 101, 257, 457, 701] -2 [5, 101, 109, 457, 641, 677, 701, 857] -1 [29, 173, 293, 977] 1 [53, 229, 733, 977] 2 [37, 197, 457, 641, 857] 3 [5, 17, 197, 229, 257, 457, 677, 733] 4 [29, 173, 293, 977] 5 [17, 29, 197, 577, 641] 6 [29, 257, 293, 401, 809] 7 [257, 577, 641, 677] 8 [5, 101, 109, 457, 641, 677, 701, 857] 9 [29, 173, 229, 293, 641, 733, 857] 10 [53, 293, 577, 701, 733] 11 [109, 197, 409, 677, 701, 809] 12 [17, 101, 257, 457, 701] 13 [5, 29, 53, 109, 173, 197, 401, 409, 857] 14 [17, 37, 229, 257, 293, 409, 457, 577, 977] 15 [37, 53, 173, 293, 577, 701, 809, 857] 16 [53, 229, 733, 977] 17 [229, 401, 677, 809] 18 [5, 37, 101, 457, 677, 701, 977] 19 [109, 229, 257, 401, 409, 677] |
def smallestChampion(E, N):
for p in prime_range(N):
if E.has_good_reduction(p):
if E.reduction(p).trace_of_frobenius() == -floor(2 * sqrt(p)):
return p # returns 'None' if no such prime < N
|
|
N = 500
spacing = 12
loA = -10
hiA = 10
loB = -10
hiB = 10
for a in range(loA, hiA):
for b in range(loB, hiB):
if (4*a*a*a + 27*b*b != 0): # not singular
E = EllipticCurve([0,0,0,a,b])
s = '(' + repr(a) + ', ' + repr(b) + ')'
print s.ljust(spacing), repr(E.j_invariant()).ljust(spacing), smallestChampion(E, N)
WARNING: Output truncated!full_output.txt (-10, -10) 69120/13 None (-10, -9) 6912000/1813 None (-10, -8) 216000/71 3 (-10, -7) 6912000/2677 None (-10, -6) 1728000/757 None (-10, -5) 276480/133 3 (-10, -4) 432000/223 457 (-10, -3) 6912000/3757 19 (-10, -2) 1728000/973 3 (-10, -1) 6912000/3973 53 (-10, 0) 1728 101 (-10, 1) 6912000/3973 3 (-10, 2) 1728000/973 None (-10, 3) 6912000/3757 11 (-10, 4) 432000/223 3 (-10, 5) 276480/133 127 (-10, 6) 1728000/757 179 (-10, 7) 6912000/2677 3 (-10, 8) 216000/71 None (-10, 9) 6912000/1813 None (-9, -10) 23328 41 (-9, -9) 6912 None (-9, -8) 46656/11 107 (-9, -7) 186624/59 None (-9, -6) 2592 None (-9, -5) 186624/83 None (-9, -4) 46656/23 None (-9, -3) 20736/11 None (-9, -2) 23328/13 None (-9, -1) 186624/107 None (-9, 0) 1728 53 (-9, 1) 186624/107 None (-9, 2) 23328/13 None (-9, 3) 20736/11 None (-9, 4) 46656/23 None (-9, 5) 186624/83 131 (-9, 6) 2592 None (-9, 7) 186624/59 None (-9, 8) 46656/11 None (-9, 9) 6912 None (-8, -10) -884736/163 11 (-8, -9) -3538944/139 None (-8, -8) 55296/5 None (-8, -7) 3538944/725 None (-8, -6) 884736/269 None (-8, -5) 3538944/1373 None (-8, -4) 221184/101 None (-8, -3) 3538944/1805 103 (-8, -2) 884736/485 19 (-8, -1) 3538944/2021 41 (-8, 0) 1728 37 (-8, 1) 3538944/2021 11 (-8, 2) 884736/485 None (-8, 3) 3538944/1805 None (-8, 4) 221184/101 None (-8, 5) 3538944/1373 None (-8, 6) 884736/269 None (-8, 7) 3538944/725 None (-8, 8) 55296/5 None ... (7, -10) 296352/509 None (7, -9) 2370816/3559 None (7, -8) 592704/775 None (7, -7) 48384/55 53 (7, -6) 296352/293 None (7, -5) 2370816/2047 None (7, -4) 592704/451 7 (7, -3) 2370816/1615 None (7, -2) 296352/185 19 (7, -1) 2370816/1399 None (7, 0) 1728 257 (7, 1) 2370816/1399 419 (7, 2) 296352/185 None (7, 3) 2370816/1615 7 (7, 4) 592704/451 None (7, 5) 2370816/2047 None (7, 6) 296352/293 None (7, 7) 48384/55 53 (7, 8) 592704/775 None (7, 9) 2370816/3559 None (8, -10) 884736/1187 5 (8, -9) 3538944/4235 41 (8, -8) 55296/59 3 (8, -7) 3538944/3371 None (8, -6) 884736/755 23 (8, -5) 3538944/2723 3 (8, -4) 221184/155 89 (8, -3) 3538944/2291 271 (8, -2) 884736/539 3 (8, -1) 3538944/2075 449 (8, 0) 1728 5 (8, 1) 3538944/2075 3 (8, 2) 884736/539 29 (8, 3) 3538944/2291 None (8, 4) 221184/155 3 (8, 5) 3538944/2723 5 (8, 6) 884736/755 None (8, 7) 3538944/3371 3 (8, 8) 55296/59 None (8, 9) 3538944/4235 41 (9, -10) 11664/13 None (9, -9) 6912/7 None (9, -8) 46656/43 41 (9, -7) 186624/157 11 (9, -6) 1296 53 (9, -5) 186624/133 None (9, -4) 46656/31 79 (9, -3) 20736/13 19 (9, -2) 11664/7 23 (9, -1) 186624/109 None (9, 0) 1728 29 (9, 1) 186624/109 151 (9, 2) 11664/7 None (9, 3) 20736/13 None (9, 4) 46656/31 11 (9, 5) 186624/133 None (9, 6) 1296 53 (9, 7) 186624/157 None (9, 8) 46656/43 41 (9, 9) 6912/7 None WARNING: Output truncated!full_output.txt (-10, -10) 69120/13 None (-10, -9) 6912000/1813 None (-10, -8) 216000/71 3 (-10, -7) 6912000/2677 None (-10, -6) 1728000/757 None (-10, -5) 276480/133 3 (-10, -4) 432000/223 457 (-10, -3) 6912000/3757 19 (-10, -2) 1728000/973 3 (-10, -1) 6912000/3973 53 (-10, 0) 1728 101 (-10, 1) 6912000/3973 3 (-10, 2) 1728000/973 None (-10, 3) 6912000/3757 11 (-10, 4) 432000/223 3 (-10, 5) 276480/133 127 (-10, 6) 1728000/757 179 (-10, 7) 6912000/2677 3 (-10, 8) 216000/71 None (-10, 9) 6912000/1813 None (-9, -10) 23328 41 (-9, -9) 6912 None (-9, -8) 46656/11 107 (-9, -7) 186624/59 None (-9, -6) 2592 None (-9, -5) 186624/83 None (-9, -4) 46656/23 None (-9, -3) 20736/11 None (-9, -2) 23328/13 None (-9, -1) 186624/107 None (-9, 0) 1728 53 (-9, 1) 186624/107 None (-9, 2) 23328/13 None (-9, 3) 20736/11 None (-9, 4) 46656/23 None (-9, 5) 186624/83 131 (-9, 6) 2592 None (-9, 7) 186624/59 None (-9, 8) 46656/11 None (-9, 9) 6912 None (-8, -10) -884736/163 11 (-8, -9) -3538944/139 None (-8, -8) 55296/5 None (-8, -7) 3538944/725 None (-8, -6) 884736/269 None (-8, -5) 3538944/1373 None (-8, -4) 221184/101 None (-8, -3) 3538944/1805 103 (-8, -2) 884736/485 19 (-8, -1) 3538944/2021 41 (-8, 0) 1728 37 (-8, 1) 3538944/2021 11 (-8, 2) 884736/485 None (-8, 3) 3538944/1805 None (-8, 4) 221184/101 None (-8, 5) 3538944/1373 None (-8, 6) 884736/269 None (-8, 7) 3538944/725 None (-8, 8) 55296/5 None ... (7, -10) 296352/509 None (7, -9) 2370816/3559 None (7, -8) 592704/775 None (7, -7) 48384/55 53 (7, -6) 296352/293 None (7, -5) 2370816/2047 None (7, -4) 592704/451 7 (7, -3) 2370816/1615 None (7, -2) 296352/185 19 (7, -1) 2370816/1399 None (7, 0) 1728 257 (7, 1) 2370816/1399 419 (7, 2) 296352/185 None (7, 3) 2370816/1615 7 (7, 4) 592704/451 None (7, 5) 2370816/2047 None (7, 6) 296352/293 None (7, 7) 48384/55 53 (7, 8) 592704/775 None (7, 9) 2370816/3559 None (8, -10) 884736/1187 5 (8, -9) 3538944/4235 41 (8, -8) 55296/59 3 (8, -7) 3538944/3371 None (8, -6) 884736/755 23 (8, -5) 3538944/2723 3 (8, -4) 221184/155 89 (8, -3) 3538944/2291 271 (8, -2) 884736/539 3 (8, -1) 3538944/2075 449 (8, 0) 1728 5 (8, 1) 3538944/2075 3 (8, 2) 884736/539 29 (8, 3) 3538944/2291 None (8, 4) 221184/155 3 (8, 5) 3538944/2723 5 (8, 6) 884736/755 None (8, 7) 3538944/3371 3 (8, 8) 55296/59 None (8, 9) 3538944/4235 41 (9, -10) 11664/13 None (9, -9) 6912/7 None (9, -8) 46656/43 41 (9, -7) 186624/157 11 (9, -6) 1296 53 (9, -5) 186624/133 None (9, -4) 46656/31 79 (9, -3) 20736/13 19 (9, -2) 11664/7 23 (9, -1) 186624/109 None (9, 0) 1728 29 (9, 1) 186624/109 151 (9, 2) 11664/7 None (9, 3) 20736/13 None (9, 4) 46656/31 11 (9, 5) 186624/133 None (9, 6) 1296 53 (9, 7) 186624/157 None (9, 8) 46656/43 41 (9, 9) 6912/7 None |
N = 1000
M = 100
all = []
for b in range(-M, M):
if b != 0:
p = smallestChampion(EllipticCurve([0,0,0,0,b]), N)
if p != None:
all.append((b, p))
allPoints = point2d(all)
show(allPoints)
|
N = 1000
M = 100
all = []
for a in range(-M, M):
if a != 0:
p = smallestChampion(EllipticCurve([0,0,0,a,0]), N)
if p != None:
all.append((a, p))
allPoints = point2d(all)
show(allPoints)
|
def isChampion(E, p):
return E.has_good_reduction(p) and E.reduction(p).trace_of_frobenius() == -floor(2 * sqrt(p))
def smallestChampion(E, N):
for p in prime_range(N):
if isChampion(E, p):
return p # returns 'None' if no such prime < N
N = 700
M = 200
count = []
# grow list to length N
for i in range(N):
count.append(0)
for b in range(-M, M):
if b != 0: # curve is nonsingular
E = EllipticCurve([0, b])
p = smallestChampion(E, N)
if p != None:
count[p] += 1
all = []
for p in prime_range(N):
all.append((p, count[p]))
allPoints = point2d(all)
show(allPoints)
|
N = 350
M = 2000
count = []
# grow list to length N
for i in range(N):
count.append(0)
for b in range(-M, M):
if b != 0: # curve is nonsingular
E = EllipticCurve([0, b])
p = smallestChampion(E, N)
if p != None:
count[p] += 1
all = []
for p in prime_range(N):
all.append((p, count[p]))
allPoints = point2d(all)
show(allPoints)
|
for i in range(len(all)):
print all[i]
(2, 0) (3, 0) (5, 0) (7, 572) (11, 0) (13, 528) (17, 0) (19, 457) (23, 0) (29, 0) (31, 391) (37, 0) (41, 0) (43, 336) (47, 0) (53, 0) (59, 0) (61, 0) (67, 274) (71, 0) (73, 248) (79, 207) (83, 0) (89, 0) (97, 172) (101, 0) (103, 138) (107, 0) (109, 0) (113, 0) (127, 0) (131, 0) (137, 0) (139, 108) (149, 0) (151, 0) (157, 92) (163, 64) (167, 0) (173, 0) (179, 0) (181, 74) (191, 0) (193, 0) (197, 0) (199, 45) (211, 47) (223, 0) (227, 0) (229, 0) (233, 0) (239, 0) (241, 35) (251, 0) (257, 0) (263, 0) (269, 0) (271, 0) (277, 0) (281, 0) (283, 0) (293, 0) (307, 41) (311, 0) (313, 29) (317, 0) (331, 0) (337, 0) (347, 0) (349, 26) (2, 0) (3, 0) (5, 0) (7, 572) (11, 0) (13, 528) (17, 0) (19, 457) (23, 0) (29, 0) (31, 391) (37, 0) (41, 0) (43, 336) (47, 0) (53, 0) (59, 0) (61, 0) (67, 274) (71, 0) (73, 248) (79, 207) (83, 0) (89, 0) (97, 172) (101, 0) (103, 138) (107, 0) (109, 0) (113, 0) (127, 0) (131, 0) (137, 0) (139, 108) (149, 0) (151, 0) (157, 92) (163, 64) (167, 0) (173, 0) (179, 0) (181, 74) (191, 0) (193, 0) (197, 0) (199, 45) (211, 47) (223, 0) (227, 0) (229, 0) (233, 0) (239, 0) (241, 35) (251, 0) (257, 0) (263, 0) (269, 0) (271, 0) (277, 0) (281, 0) (283, 0) (293, 0) (307, 41) (311, 0) (313, 29) (317, 0) (331, 0) (337, 0) (347, 0) (349, 26) |
def isChampion(E, p):
return E.has_good_reduction(p) and E.reduction(p).trace_of_frobenius() == -floor(2 * sqrt(p))
N = 700
M = 100
all = []
for b in range(1, M):
E = EllipticCurve([0, b])
for p in prime_range(1, N):
if p % 6 == 1 and isChampion(E, p):
all.append((b, p))
allPoints = point2d(all)
show(allPoints)
|
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