F.<a> = NumberField(x^2 - x - 1)
E = EllipticCurve([1,a+1,a,a,0])
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P = sum([[I for I in P if I.is_prime()] for _,P in F.ideals_of_bdd_norm(1000).iteritems()], [])
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v = []
aplist = []
for p in P[:50]:
try:
k = p.residue_field()
Np = E.change_ring(k).cardinality()
ap = k.cardinality() + 1 - Np
aplist.append(ap)
v.append(ap/(2*math.sqrt(k.cardinality())))
except Exception, msg:
print msg
Invariants [1, 20, 19, 19, 0] define a singular curve. Invariants [1, 20, 19, 19, 0] define a singular curve. |
aplist
[-3, -2, 2, -4, 4, 4, -4, -2, -2, 8, -6, -6, 2, -4, 12, -2, 6, -8, 0, 16, 0, 10, -6, -10, 6, -10, 6, 4, -20, -20, 4, -10, 6, 16, 8, -6, -12, 4, 22, -10, 16, 0, 24, 16, -4, -12, -26, 6, 0] [-3, -2, 2, -4, 4, 4, -4, -2, -2, 8, -6, -6, 2, -4, 12, -2, 6, -8, 0, 16, 0, 10, -6, -10, 6, -10, 6, 4, -20, -20, 4, -10, 6, 16, 8, -6, -12, 4, 22, -10, 16, 0, 24, 16, -4, -12, -26, 6, 0] |
[p.gens_reduced()[0] for p in P[:50]]
[2, 2*a - 1, 3, 3*a - 1, 3*a - 2, -4*a + 1, -4*a + 3, -a + 6, a + 5, 5*a - 3, 5*a - 2, a - 7, a + 6, 7, 7*a - 5, 7*a - 2, 7*a - 4, 7*a - 3, a - 9, a + 8, -8*a + 3, -8*a + 5, a - 10, a + 9, 9*a - 5, 9*a - 4, a - 11, a + 10, a - 12, a + 11, 2*a + 11, -2*a + 13, -11*a + 4, -11*a + 7, 11*a - 6, 11*a - 5, 13, -12*a + 5, 12*a - 7, a - 14, a + 13, 2*a + 13, -2*a + 15, 3*a + 13, -3*a + 16, 13*a - 7, 13*a - 6, 3*a - 17, -3*a - 14, a - 16] [2, 2*a - 1, 3, 3*a - 1, 3*a - 2, -4*a + 1, -4*a + 3, -a + 6, a + 5, 5*a - 3, 5*a - 2, a - 7, a + 6, 7, 7*a - 5, 7*a - 2, 7*a - 4, 7*a - 3, a - 9, a + 8, -8*a + 3, -8*a + 5, a - 10, a + 9, 9*a - 5, 9*a - 4, a - 11, a + 10, a - 12, a + 11, 2*a + 11, -2*a + 13, -11*a + 4, -11*a + 7, 11*a - 6, 11*a - 5, 13, -12*a + 5, 12*a - 7, a - 14, a + 13, 2*a + 13, -2*a + 15, 3*a + 13, -3*a + 16, 13*a - 7, 13*a - 6, 3*a - 17, -3*a - 14, a - 16] |
v
[-0.75, -0.44721359549995793, 0.33333333333333331, -0.60302268915552726, 0.60302268915552726, 0.45883146774112349, -0.45883146774112349, -0.18569533817705186, -0.18569533817705186, 0.71842120810709964, -0.46852128566581819, -0.46852128566581819, 0.14285714285714285, -0.26037782196164777, 0.78113346588494326, -0.12803687993289598, 0.38411063979868793, -0.4747126632775413, 0.0, 0.90007032074081916, 0.0, 0.52999894000318004, -0.31799936400190804, -0.49751859510499458, 0.29851115706299675, -0.47891314261057566, 0.28734788556634538, 0.17474081133220759, -0.8737040566610379, -0.84818892967997095, 0.16963778593599418, -0.40961596025952024, 0.24576957615571215, 0.65103076701692753, 0.32551538350846376, -0.23076923076923078, -0.44846105565116151, 0.14948701855038718, 0.81762356087188293, -0.37164707312358319, 0.5788596842339373, 0.0, 0.85065744601000304, 0.56710496400666865, -0.13768567816430285, -0.41305703449290854, -0.85906418059161671, 0.19824558013652693, 0.0] [-0.75, -0.44721359549995793, 0.33333333333333331, -0.60302268915552726, 0.60302268915552726, 0.45883146774112349, -0.45883146774112349, -0.18569533817705186, -0.18569533817705186, 0.71842120810709964, -0.46852128566581819, -0.46852128566581819, 0.14285714285714285, -0.26037782196164777, 0.78113346588494326, -0.12803687993289598, 0.38411063979868793, -0.4747126632775413, 0.0, 0.90007032074081916, 0.0, 0.52999894000318004, -0.31799936400190804, -0.49751859510499458, 0.29851115706299675, -0.47891314261057566, 0.28734788556634538, 0.17474081133220759, -0.8737040566610379, -0.84818892967997095, 0.16963778593599418, -0.40961596025952024, 0.24576957615571215, 0.65103076701692753, 0.32551538350846376, -0.23076923076923078, -0.44846105565116151, 0.14948701855038718, 0.81762356087188293, -0.37164707312358319, 0.5788596842339373, 0.0, 0.85065744601000304, 0.56710496400666865, -0.13768567816430285, -0.41305703449290854, -0.85906418059161671, 0.19824558013652693, 0.0] |
stats.TimeSeries(v).plot_histogram(20)
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