Sharedhöhere_Verzweigungsgruppen / Example 3.6. - Conductors .sagewsOpen in CoCalc
Wild Conductor of Elliptic Curves
#E1: y^2 = x^3 + 2
E1 = EllipticCurve([0,0,0,0,2]); E1
L1.<a1> = E1.division_field(3)
G1 = L1.galois_group()
G1.ramification_breaks(L1.primes_above(2)[0])
G1.inertia_group(L1.primes_above(2)[0]).order()
Elliptic Curve defined by y^2 = x^3 + 2 over Rational Field {2, -1} 2
#So the cardinalities of the ramification subgroups of L1/QQ_2 is given by: g0 = g1 = g2 = 2 and gi = 1 for i > 2
#and we can caluclate the wild part of the conductor for E1. delta1 = 2* (g1/g0 + g2/g0)
delta1 = 2 *(1+1); delta1
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#E2: y^2 + 2*y = x^3 + 2
E2= EllipticCurve([0,0,2,0,2]); E2
L2.<a2> = E2.division_field(3)
G2 = L2.galois_group()
G2.ramification_breaks(L2.primes_above(2)[0])
G2.inertia_group(L2.primes_above(2)[0]).order()
Elliptic Curve defined by y^2 + 2*y = x^3 + 2 over Rational Field {0, 3, -1} 6
##So the cardinalities of the ramification subgroups of L2/QQ_2 is given by: g0 = 6,  g1 = g2 = g3 = 2 and gi = 1 for i > 3
#and we can caluclate the wild part of the conductor for E2. delta2 = 2* (g1/g0 + g2/g0+g3/g0)
delta2 = 2*(1/3 +1/3 +1/3); delta2
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#E3: y^2 + 2*x*y = x^3 + 2*x + 2
E3 = EllipticCurve([2,0,0,2,2]); E3
#The division field L3.<a3> = E3.division_field(3) is "too big" for Galois group computations, so we use a weak padic Galois extension (of the mclf package).
import sys
sys.path.append('../MCLF')
from mclf import *

#First we find a polynomial with splitting field equal to the field extension generated by the 3-torsion points.
g = E3.division_polynomial(3)
#The roots of g are the x-coordinates of the 3-torsion points. We caluclate the the resultant of g and the polynomial defining E3 in order to construct a field that also contains the y-coordinates of the 3-torsion points.
R.<y> = QQ[]
S.<x> = R[]
G = g(x)
F = y^2  + 2*x*y - x^3 - 2*x - 2
f = F.resultant(G); f

#Construction of the Weak p-adic Galois Extension with defining polynomial f
v_2 = QQ.valuation(2)
Q2 = FakepAdicCompletion(QQ, v_2)
L3 = WeakPadicGaloisExtension(Q2, f.monic()); L3
L3.ramification_filtration()
Elliptic Curve defined by y^2 + 2*x*y = x^3 + 2*x + 2 over Rational Field 27*y^8 - 72*y^7 + 568*y^6 - 528*y^5 - 984*y^4 + 4768*y^3 + 2400*y^2 - 7872*y - 5840 2-adic completion of Number Field in pi24 with defining polynomial x^24 + 52*x^21 + 28*x^18 + 24*x^15 + 24*x^12 + 24*x^9 + 124*x^6 + 80*x^3 + 22 as weak Galois extension of 2-adic completion of Rational Field [(0, 24), (5, 8), (15, 2)]
##So the cardinalities of the ramification subgroups of L3/QQ_3 is given by: g0 = 24,  g1 = ... = g5 = 8, g6 = ... = g15 = 2 and gi = 1 for i > 15
#and we can caluclate the wild part of the conductor for E3.
delta3 = 2* (5*1/3 + 10*1/12); delta3
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