# This file lists all examples in our tables that do not have a BelyiMap defined over the ModuliField.
#
# Examples that have the same ModuliField and the same set of Obstruction primes are grouped together.
#
# Note that the ModuliFields in this file are Galois over Q, and that to match the given conic
# to the corresponding entry in the file BelyiMap, you may have to take a conjugate over Q
# (comparing the real obstructions (the primes at infinity) will tell you which conjugate to take).
#
# Notation:
#
# Obstruction[ID]  := [ ModuliField, Obstruction Primes, Obstruction Conic]; # Decomposition
#
# We also listed the smallest cyclotomic fields for which a BelyiMap defined over that field
# can be given.



Obstruction["B12"] := ["Q", [infinity, 7],  7*X^2 +   Y^2 + Z^2 = 0]; # "S11(deg 4)"
	# Has a model over Q(zeta_4)

Obstruction["C6"]  := ["Q", [infinity, 2],    X^2 +   Y^2 + Z^2 = 0]; # "S18(deg 4)"
	# Has a model over Q(zeta_4)

Obstruction["C30"] := ["Q", [infinity, 5],  5*X^2 + 2*Y^2 + Z^2 = 0]; # "Indecomposable"
Obstruction["D45"] := ["Q", [infinity, 5],  5*X^2 + 2*Y^2 + Z^2 = 0]; # "C30(deg 2)"
	# Have a model over Q(zeta_3)

Obstruction["F1"]  := ["Q(sqrt(2))", [infinity, 3],  (1-2^(1/2))*X^2 + 3*Y^2 + Z^2 = 0]; # "Indecomposable"
	# Has a model over a subfield of Q(zeta_16), and over a subfield of Q(zeta_24)

Obstruction["F4"]  := ["Q(sqrt(2))", [infinity, infinity],  X^2+Y^2+Z^2 = 0]; # "Indecomposable"
Obstruction["F6"]  := ["Q(sqrt(2))", [infinity, infinity],  X^2+Y^2+Z^2 = 0]; # "F4(deg 2)"
	# Have a model over Q(zeta_8)

Obstruction["F11"] := ["Q(sqrt(5))", [infinity, 5], (2+5^(1/2))*X^2 + 5^(1/2)*Y^2 + Z^2 = 0]; # "Indecomposable"
	# Has a model over Q(zeta_5)

Obstruction["H1"]  := ["Q(x^3-3*x-1) = Q(Re(zeta_9))",     [infinity, infinity], X^2 + Y^2 - RootOf(x^3-3*x-1) * Z^2 = 0]; # "Indecomposable"
	# Has a model over Q(zeta_9)

Obstruction["H10"] := ["Q(x^3-x^2-2*x+1) = Q(Re(zeta_7))", [infinity, infinity], X^2 + Y^2 + RootOf( x^3-x^2-2*x+1 ) * Z^2 = 0]; # "Indecomposable"
Obstruction["H11"] := ["Q(x^3-x^2-2*x+1) = Q(Re(zeta_7))", [infinity, infinity], X^2 + Y^2 + RootOf( x^3-x^2-2*x+1 ) * Z^2 = 0]; # "Indecomposable"
Obstruction["H12"] := ["Q(x^3-x^2-2*x+1) = Q(Re(zeta_7))", [infinity, infinity], X^2 + Y^2 + RootOf( x^3-x^2-2*x+1 ) * Z^2 = 0]; # "Indecomposable"
Obstruction["H13"] := ["Q(x^3-x^2-2*x+1) = Q(Re(zeta_7))", [infinity, infinity], X^2 + Y^2 + RootOf( x^3-x^2-2*x+1 ) * Z^2 = 0]; # "H12(deg 2)"
Obstruction["H14"] := ["Q(x^3-x^2-2*x+1) = Q(Re(zeta_7))", [infinity, infinity], X^2 + Y^2 + RootOf( x^3-x^2-2*x+1 ) * Z^2 = 0]; # "H10(deg 2)"
	# Have a model over Q(zeta_7)


# Observation #1:   F1, F11, H1, and H10-H14 each have an interesting feature, namely:
#
#		    *) they have n>1 dessins, all of them are real (i.e. invariant under complex conjugation)
#		    *) they have 1 dessin that allows a Belyi map over the reals, and n-1 that do not.
#
# So, among these "real" dessins, some are more real than others.


# Observation #2:   The Belyi maps Tables A-J with a ModuliField of degree >= 3 that have an obstruction
#                   are precisely those that have a ModuliField with cyclic Galois group.
#                   They are: H1, H10, H11, H12, H13, H14.  To indicate the cyclic Galois group we
#                   rewrote Q(x^3-3*x-1) as Q(Re(zeta_9)), and Q(x^3-x^2-2*x+1) as Q(Re(zeta_7)).