CoCalc -- profiletwists.sagews

Project: Harvard Workshop: Distribution of modular symbols and L-values

Path: code / alex / psage / profiletwists.sagews

Views: ¹⁸⁴⁶⁰⁴

attach("twistedlseries.sage")

psage/modform/hilbert/sqrt5/sqrt5.py:852: DeprecationWarning: python.pxi is deprecated, use "from cpython cimport *" instead
See http://trac.sagemath.org/20158 for details.
  from tables import primes_of_bounded_norm
psage/number_fields/__init__.py:1: DeprecationWarning: import "cysignals/signals.pxi" instead of "sage/ext/interrupt.pxi"
See http://trac.sagemath.org/20002 for details.
  import sqrt5

E = EllipticCurve(F, [1, a + 1, a, a, 0])
E
L = LSeriesEllipticCurveSqrt5(E)
chi = NFChar([F.prime_above(2), F.prime_above(31)], [1,1], 3)
chi.conductor()
#LEchi = L.twist(chi)
#LEchi.conductor()
#LEchi.check_functional_equation(1.2)
#LEchi.check_functional_equation(1.1)
#factor(LEchi.conductor())

Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1
124

#%timeit(repeat=1,number=1) 
vals,nvals,chis = value_search(E,3,100)
print chis
print vals
print nvals

Fractional ideal (7) 49 Fractional ideal (7) [1]
Fractional ideal (-8*a + 2) 76 (Fractional ideal (2)) * (Fractional ideal (-4*a + 1)) [1, 2]
Fractional ideal (8*a - 6) 76 (Fractional ideal (2)) * (Fractional ideal (4*a - 3)) [1, 1]
raw value 4.23997384119518e-16 + 4.63884971589829e-16*I
normalised 6.83594740282864e-17 - 3.75785568287707e-16*I
algdeprts [(0.000000000000000, 1)]
raw value 1.77476876452315 - 1.95759988981642*I
normalised 1.00000000000000 + 1.73205080756888*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
raw value 8.57233239493546e-17 - 1.06432925047696e-16*I
normalised -9.65405943620137e-17 - 3.71441181259954e-17*I
algdeprts [(0.000000000000000, 1)]
(Hecke character of modulus Fractional ideal (7) and order 3, Hecke character of modulus Fractional ideal (-8*a + 2) and order 3, Hecke character of modulus Fractional ideal (8*a - 6) and order 3)
(4.23997384119518e-16 + 4.63884971589829e-16*I, 1.77476876452315 - 1.95759988981642*I, 8.57233239493546e-17 - 1.06432925047696e-16*I)
(6.83594740282864e-17 - 3.75785568287707e-16*I, 1.00000000000000 + 1.73205080756888*I, -9.65405943620137e-17 - 3.71441181259954e-17*I)

%timeit(repeat=1,number=1) L._primes_above(499)
%timeit(repeat=1,number=1) L._primes_above(499)
%timeit(repeat=1,number=1) L._primes_above(499)

loops, best of 1: 13.1 µs per loop
loops, best of 1: 5.96 µs per loop
loops, best of 1: 5.96 µs per loop


[chi.conductor() for chi in chis]

Fractional ideal (7) 49 Fractional ideal (7) [1]
raw value 4.23997384119518e-16 + 4.63884971589829e-16*I
normalised 6.83594740282864e-17 - 3.75785568287707e-16*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-8*a + 2) 76 (Fractional ideal (2)) * (Fractional ideal (-4*a + 1)) [1, 2]
raw value 1.77476876452315 - 1.95759988981642*I
normalised 1.00000000000000 + 1.73205080756888*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (8*a - 6) 76 (Fractional ideal (2)) * (Fractional ideal (4*a - 3)) [1, 2]
raw value 3.96635225811821e-16 + 8.65558293069396e-16*I
normalised -7.15561925897464e-16 + 8.55135113864284e-17*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (2*a + 11) 139 Fractional ideal (2*a + 11) [1]
raw value 1.94848534487254 - 0.144575522568633*I
normalised 0.999999999999995 + 1.73205080756887*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (-2*a + 13) 139 Fractional ideal (-2*a + 13) [1]
raw value -1.09944874774612 - 1.61515004627698*I
normalised 0.999999999999997 - 1.73205080756887*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (11*a - 6) 151 Fractional ideal (11*a - 6) [1]
raw value 1.28139749191622 - 1.36826182656356*I
normalised -1.99999999999999 - 1.54228076864596e-16*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (11*a - 5) 151 Fractional ideal (11*a - 5) [1]
raw value 3.84419247574867 - 4.10478547969069*I
normalised -5.99999999999998 - 4.31838615220870e-15*I
algdeprts [(-6.00000000000000, 1)]
Fractional ideal (13) 169 Fractional ideal (13) [1]
raw value 1.54579562381928 - 0.866227961755150*I
normalised 2.00000000000000 - 6.16912307458385e-16*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (3*a + 13) 199 Fractional ideal (3*a + 13) [1]
raw value 1.29290702755217 + 0.997438946405499*I
normalised 2.00000000000001 - 6.94026345890684e-15*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (-3*a + 16) 199 Fractional ideal (-3*a + 16) [1]
raw value 1.51026098008723 - 0.620970857388854*I
normalised 0.999999999999999 - 1.73205080756889*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (14*a - 8) 244 (Fractional ideal (2)) * (Fractional ideal (7*a - 4)) [1, 1]
raw value 0.0314883405142121 - 1.47435757599598*I
normalised -2.00000000000000 + 9.15729206383541e-16*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (14*a - 6) 244 (Fractional ideal (2)) * (Fractional ideal (7*a - 3)) [1, 2]
raw value 1.29257528533167 + 0.709909085189665*I
normalised -0.999999999999998 + 1.73205080756888*I
algdeprts [(-1.00000000000000 - 1.73205080756888*I, 1), (-1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (-16*a + 4) 304 (Fractional ideal (2))^2 * (Fractional ideal (-4*a + 1)) [1, 2]
raw value 1.77476876452315 - 1.95759988981642*I
normalised 1.00000000000000 + 1.73205080756888*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (16*a - 12) 304 (Fractional ideal (2))^2 * (Fractional ideal (4*a - 3)) [1, 2]
raw value 3.96635225811821e-16 + 8.65558293069396e-16*I
normalised -7.15561925897464e-16 + 8.55135113864284e-17*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-16*a + 6) 316 (Fractional ideal (2)) * (Fractional ideal (-8*a + 3)) [1, 1]
raw value 1.28951584857673 + 0.127923516106712*I
normalised 0.999999999999997 + 1.73205080756888*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (-16*a + 10) 316 (Fractional ideal (2)) * (Fractional ideal (-8*a + 5)) [1, 1]
raw value 0.533972909598525 + 1.18071524150345*I
normalised -1.00000000000000 + 1.73205080756888*I
algdeprts [(-1.00000000000000 - 1.73205080756888*I, 1), (-1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (3*a - 20) 331 Fractional ideal (3*a - 20) [1]
raw value 5.51133057385612 - 3.11499430386439*I
normalised 9.99999999999998 - 1.85073692237516e-15*I
algdeprts [(10.0000000000000, 1)]
Fractional ideal (-3*a - 17) 331 Fractional ideal (-3*a - 17) [1]
raw value 0.0116002174275343 - 1.26608988750909*I
normalised 0.999999999999996 - 1.73205080756887*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (19) 361 (Fractional ideal (4*a - 3)) * (Fractional ideal (-4*a + 1)) [1, 1]
raw value 3.31254869309870 - 1.50202417497642*I
normalised -2.99999999999999 + 5.19615242270663*I
algdeprts [(-3.00000000000000 - 5.19615242270663*I, 1), (-3.00000000000000 + 5.19615242270663*I, 1)]
Fractional ideal (2*a - 22) 436 (Fractional ideal (2)) * (Fractional ideal (a - 11)) [1, 1]
raw value 0.105811424643225 + 3.30790252782655*I
normalised 6.00000000000002 - 9.71636884246957e-15*I
algdeprts [(6.00000000000000, 1)]
Fractional ideal (2*a + 20) 436 (Fractional ideal (2)) * (Fractional ideal (a + 10)) [1, 2]
raw value 0.937273970006314 + 0.581862215221642*I
normalised -0.999999999999999 + 1.73205080756888*I
algdeprts [(-1.00000000000000 - 1.73205080756888*I, 1), (-1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (3*a + 22) 541 Fractional ideal (3*a + 22) [1]
 0.937273970006314 + 0.581862215221642*I
normalised -0.999999999999999 + 1.73205080756888*I
algdeprts [(-1.00000000000000 - 1.73205080756888*I, 1), (-1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (3*a + 22) 541 Fractional ideal (3*a + 22) [1]
raw value 0.220342943690387 - 0.965548904601734*I
normalised -0.999999999999960 - 1.73205080756884*I
algdeprts [(-1.00000000000000 - 1.73205080756888*I, 1), (-1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (-3*a + 25) 541 Fractional ideal (-3*a + 25) [1]
raw value 5.67816811095919 - 1.75171119312209*I
normalised 5.99999999999994 - 10.3923048454130*I
algdeprts [(6.00000000000000 - 10.3923048454133*I, 1), (6.00000000000000 + 10.3923048454133*I, 1)]
Fractional ideal (-5*a - 23) 619 Fractional ideal (-5*a - 23) [1]
raw value 2.97938352704803e-17 + 3.43681605226057e-16*I
normalised 7.44849374231626e-16 + 2.21641516343011e-17*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-5*a + 28) 619 Fractional ideal (-5*a + 28) [1]
raw value 0.850102587652341 + 0.366833047835574*I
normalised 0.999999999999995 - 1.73205080756885*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (23*a - 12) 661 Fractional ideal (23*a - 12) [1]
raw value -1.05030706605215 - 1.45187533489213*I
normalised -1.99999999999996 - 3.46410161513771*I
algdeprts [(-2.00000000000000 - 3.46410161513775*I, 1), (-2.00000000000000 + 3.46410161513775*I, 1)]
Fractional ideal (23*a - 11) 661 Fractional ideal (23*a - 11) [1]
raw value 2.67377168425605 - 0.275482400294101*I
normalised 5.99999999999991 + 2.15919307610435e-14*I
algdeprts [(6.00000000000000, 1)]
Fractional ideal (4*a + 25) 709 Fractional ideal (4*a + 25) [1]
raw value 0.407925749775957 + 0.762902270422673*I
normalised -0.999999999999962 - 1.73205080756885*I
algdeprts [(-1.00000000000000 - 1.73205080756888*I, 1), (-1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (-4*a + 29) 709 Fractional ideal (-4*a + 29) [1]
Interrupting PARI/GP interpreter...

Error in lines 1-1
Traceback (most recent call last):
  File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute
    exec compile(block+'\n', '', 'single') in namespace, locals
  File "", line 1, in <module>
  File "<string>", line 150, in value_search
  File "psage/lseries/eulerprod.py", line 1507, in check_functional_equation
    return self._function(prec=prec,T=T).check_functional_equation(T)
  File "sage/misc/cachefunc.pyx", line 2038, in sage.misc.cachefunc.CachedMethodCaller.__call__ (/projects/sage/sage-7.5/src/build/cythonized/sage/misc/cachefunc.c:10792)
    w = self._instance_call(*args, **kwds)
  File "sage/misc/cachefunc.pyx", line 1914, in sage.misc.cachefunc.CachedMethodCaller._instance_call (/projects/sage/sage-7.5/src/build/cythonized/sage/misc/cachefunc.c:10238)
    return self.f(self._instance, *args, **kwds)
  File "psage/lseries/eulerprod.py", line 1408, in _function
    return self._dokchitser(prec, eps, T=T)
  File "psage/lseries/eulerprod.py", line 1495, in _dokchitser
    fe = L.check_functional_equation()
  File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/lfunctions/dokchitser.py", line 561, in check_functional_equation
    z = self.gp().eval('checkfeq(%s)'%T).replace(' ','')
  File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/interfaces/expect.py", line 1297, in eval
    for L in code.split('\n') if L != ''])
  File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/interfaces/gp.py", line 441, in _eval_line
    wait_for_prompt=wait_for_prompt)
  File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/interfaces/expect.py", line 972, in _eval_line
    self._keyboard_interrupt()
  File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/interfaces/expect.py", line 994, in _keyboard_interrupt
    raise KeyboardInterrupt("Ctrl-c pressed while running %s"%self)
KeyboardInterrupt: Ctrl-c pressed while running PARI/GP interpreter

LEchi = L.twist(NFChar([F.primes_above(139)[0]],[1],3),epsilon='solve')
LEchi.anlist(100)
LEchi.check_functional_equation(1.2)
LEchi.check_functional_equation(1.1)

[0, 1, -0.000000000000000, -0.000000000000000, 1.50000000000000 - 2.59807621135332*I, 1.00000000000000 - 1.73205080756888*I, 0.000000000000000, -0.000000000000000, -0.000000000000000, -1.00000000000000 - 1.73205080756888*I, -0.000000000000000, 6.00000000000000 - 3.46410161513775*I, -0.000000000000000, 0, 0.000000000000000, -0.000000000000000, -2.50000000000000 - 4.33012701892219*I, 0, 0.000000000000000, -6.00000000000000 + 3.46410161513775*I, -3.00000000000000 - 5.19615242270663*I, 0.000000000000000, -0.000000000000000, 0, 0.000000000000000, 0.500000000000000 + 0.866025403784438*I, 0.000000000000000, -0.000000000000000, -0.000000000000000, 2.00000000000000 + 3.46410161513775*I, 0.000000000000000, -5.00000000000000 - 6.92820323027551*I, -0.000000000000000, -0.000000000000000, 0.000000000000000, -0.000000000000000, -6.00000000000000 + 3.99680288865056e-15*I, 0, 0.000000000000000, 0.000000000000000, -0.000000000000000, -3.00000000000000 + 5.19615242270663*I, 0.000000000000000, 0, -5.32907051820075e-15 - 20.7846096908265*I, -4.00000000000000 + 2.44249065417534e-15*I, 0.000000000000000, 0, 0.000000000000000, -1.00000000000000 - 1.73205080756888*I, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0, 0.000000000000000, -4.44089209850063e-15 - 13.8564064605510*I, 0.000000000000000, 0.000000000000000, 0.000000000000000, -4.00000000000000 - 13.8564064605510*I, 0.000000000000000, 7.00000000000000 - 1.73205080756888*I, 0.000000000000000, 0.000000000000000, -2.99999999999999 - 8.88178419700125e-16*I, 0.000000000000000, 0.000000000000000, 0, 0.000000000000000, 0.000000000000000, 0.000000000000000, 4.00000000000000 - 6.92820323027551*I, 0.000000000000000, 0, 0.000000000000000, 0.000000000000000, 5.32907051820075e-15 + 20.7846096908265*I, -0.000000000000000, 0.000000000000000, -8.00000000000000 + 13.8564064605510*I, -10.0000000000000 + 5.32907051820075e-15*I, 2.50000000000000 - 4.33012701892219*I, 0.000000000000000, 0, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, -0.000000000000000, -2.00000000000000 + 3.46410161513776*I, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 4.44089209850063e-15 + 13.8564064605510*I, 0.000000000000000, 0, 0.000000000000000, -12.0000000000000 - 6.92820323027550*I, 3.00000000000000 - 2.22044604925031e-15*I]
2.77555756156289e-17 + 2.77555756156289e-17*I
-8.33777491493493e-14 - 2.03448369262560e-14*I

import cProfile
#L = LSeriesEllipticCurveSqrt5(E)
#LEchi = L.twist(NFChar([F.primes_above(709)[1]],[1],3),epsilon='solve')
t = cputime()
cProfile.runctx("value_search(E,3,200)",None, locals(), filename="timing_data/profile_test", sort='cumulative')
print cputime(t)

200 {1: [Fractional ideal (1)], 2: [], 3: [], 4: [Fractional ideal (2)], 5: [Fractional ideal (-2*a + 1)], 6: [], 7: [], 8: [], 9: [Fractional ideal (3)], 10: [], 11: [Fractional ideal (-3*a + 1), Fractional ideal (-3*a + 2)], 12: [], 13: [], 14: [], 15: [], 16: [Fractional ideal (4)], 17: [], 18: [], 19: [Fractional ideal (-4*a + 1), Fractional ideal (4*a - 3)], 20: [Fractional ideal (-4*a + 2)], 21: [], 22: [], 23: [], 24: [], 25: [Fractional ideal (5)], 26: [], 27: [], 28: [], 29: [Fractional ideal (a - 6), Fractional ideal (a + 5)], 30: [], 31: [Fractional ideal (5*a - 3), Fractional ideal (5*a - 2)], 32: [], 33: [], 34: [], 35: [], 36: [Fractional ideal (6)], 37: [], 38: [], 39: [], 40: [], 41: [Fractional ideal (a - 7), Fractional ideal (a + 6)], 42: [], 43: [], 44: [Fractional ideal (-6*a + 2), Fractional ideal (-6*a + 4)], 45: [Fractional ideal (-6*a + 3)], 46: [], 47: [], 48: [], 49: [Fractional ideal (7)], 50: [], 51: [], 52: [], 53: [], 54: [], 55: [Fractional ideal (a + 7), Fractional ideal (-a + 8)], 56: [], 57: [], 58: [], 59: [Fractional ideal (7*a - 5), Fractional ideal (7*a - 2)], 60: [], 61: [Fractional ideal (7*a - 4), Fractional ideal (7*a - 3)], 62: [], 63: [], 64: [Fractional ideal (8)], 65: [], 66: [], 67: [], 68: [], 69: [], 70: [], 71: [Fractional ideal (a - 9), Fractional ideal (a + 8)], 72: [], 73: [], 74: [], 75: [], 76: [Fractional ideal (-8*a + 2), Fractional ideal (8*a - 6)], 77: [], 78: [], 79: [Fractional ideal (-8*a + 3), Fractional ideal (-8*a + 5)], 80: [Fractional ideal (-8*a + 4)], 81: [Fractional ideal (9)], 82: [], 83: [], 84: [], 85: [], 86: [], 87: [], 88: [], 89: [Fractional ideal (a - 10), Fractional ideal (a + 9)], 90: [], 91: [], 92: [], 93: [], 94: [], 95: [Fractional ideal (2*a + 9), Fractional ideal (-2*a + 11)], 96: [], 97: [], 98: [], 99: [Fractional ideal (-9*a + 3), Fractional ideal (-9*a + 6)], 100: [Fractional ideal (10)], 101: [Fractional ideal (9*a - 5), Fractional ideal (9*a - 4)], 102: [], 103: [], 104: [], 105: [], 106: [], 107: [], 108: [], 109: [Fractional ideal (a - 11), Fractional ideal (a + 10)], 110: [], 111: [], 112: [], 113: [], 114: [], 115: [], 116: [Fractional ideal (2*a - 12), Fractional ideal (2*a + 10)], 117: [], 118: [], 119: [], 120: [], 121: [Fractional ideal (10*a - 7), Fractional ideal (11), Fractional ideal (10*a - 3)], 122: [], 123: [], 124: [Fractional ideal (10*a - 6), Fractional ideal (10*a - 4)], 125: [Fractional ideal (-10*a + 5)], 126: [], 127: [], 128: [], 129: [], 130: [], 131: [Fractional ideal (a - 12), Fractional ideal (a + 11)], 132: [], 133: [], 134: [], 135: [], 136: [], 137: [], 138: [], 139: [Fractional ideal (2*a + 11), Fractional ideal (-2*a + 13)], 140: [], 141: [], 142: [], 143: [], 144: [Fractional ideal (12)], 145: [Fractional ideal (-11*a + 8), Fractional ideal (-11*a + 3)], 146: [], 147: [], 148: [], 149: [Fractional ideal (-11*a + 4), Fractional ideal (-11*a + 7)], 150: [], 151: [Fractional ideal (11*a - 6), Fractional ideal (11*a - 5)], 152: [], 153: [], 154: [], 155: [Fractional ideal (a - 13), Fractional ideal (a + 12)], 156: [], 157: [], 158: [], 159: [], 160: [], 161: [], 162: [], 163: [], 164: [Fractional ideal (2*a - 14), Fractional ideal (2*a + 12)], 165: [], 166: [], 167: [], 168: [], 169: [Fractional ideal (13)], 170: [], 171: [Fractional ideal (-12*a + 3), Fractional ideal (12*a - 9)], 172: [], 173: [], 174: [], 175: [], 176: [Fractional ideal (-12*a + 4), Fractional ideal (-12*a + 8)], 177: [], 178: [], 179: [Fractional ideal (-12*a + 5), Fractional ideal (12*a - 7)], 180: [Fractional ideal (-12*a + 6)], 181: [Fractional ideal (a - 14), Fractional ideal (a + 13)], 182: [], 183: [], 184: [], 185: [], 186: [], 187: [], 188: [], 189: [], 190: [], 191: [Fractional ideal (2*a + 13), Fractional ideal (-2*a + 15)], 192: [], 193: [], 194: [], 195: [], 196: [Fractional ideal (14)], 197: [], 198: [], 199: [Fractional ideal (3*a + 13), Fractional ideal (-3*a + 16)], 200: []}
Fractional ideal (7) 49 Fractional ideal (7) [1]
raw value -3.64166002914050e-16 + 4.51452061713266e-16*I
normalised -8.43622242952448e-17 - 3.42270203365127e-16*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (-8*a + 2) 76 (Fractional ideal (2)) * (Fractional ideal (-4*a + 1)) [1, 2]
raw value 1.77476876452315 - 1.95759988981642*I
normalised 1.00000000000000 + 1.73205080756888*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (8*a - 6) 76 (Fractional ideal (2)) * (Fractional ideal (4*a - 3)) [1, 2]
raw value 3.60367665679402e-16 + 7.85460562254218e-16*I
normalised -6.49445722875965e-16 + 7.79148223906000e-17*I
algdeprts [(0.000000000000000, 1)]
Fractional ideal (2*a + 11) 139 Fractional ideal (2*a + 11) [1]
raw value 1.94848534487254 - 0.144575522568633*I
normalised 0.999999999999995 + 1.73205080756887*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (-2*a + 13) 139 Fractional ideal (-2*a + 13) [1]
raw value -1.09944874774612 - 1.61515004627698*I
normalised 0.999999999999997 - 1.73205080756887*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
Fractional ideal (11*a - 6) 151 Fractional ideal (11*a - 6) [1]
raw value 1.28139749191622 - 1.36826182656356*I
normalised -1.99999999999999 - 1.54228076864596e-16*I
algdeprts [(-2.00000000000000, 1)]
Fractional ideal (11*a - 5) 151 Fractional ideal (11*a - 5) [1]
raw value 3.84419247574867 - 4.10478547969069*I
normalised -5.99999999999998 - 4.31838615220870e-15*I
algdeprts [(-6.00000000000000, 1)]
Fractional ideal (13) 169 Fractional ideal (13) [1]
raw value 1.54579562381928 + 0.866227961755149*I
normalised 2.00000000000000 - 1.85073692237516e-15*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (3*a + 13) 199 Fractional ideal (3*a + 13) [1]
raw value 1.29290702755217 + 0.997438946405499*I
normalised 2.00000000000001 - 6.94026345890684e-15*I
algdeprts [(2.00000000000000, 1)]
Fractional ideal (-3*a + 16) 199 Fractional ideal (-3*a + 16) [1]
raw value 1.51026098008723 - 0.620970857388854*I
normalised 0.999999999999999 - 1.73205080756889*I
algdeprts [(1.00000000000000 - 1.73205080756888*I, 1), (1.00000000000000 + 1.73205080756888*I, 1)]
111.584