Let L be the function field of X1(17) over Q.
For f in L, deg(f) is the index [L:Q(f)].
Denote Gal(f) as the Galois group of the normal closure of this extension.
L has, up to diamond action, and up to Mobius transformations f -> (a*f+b)/(c*f+d),
only three functions of degree 4.
We can write L = Q(x,y) where x,y satisfy the equation from Drew's website:
Fxy := x^4*y-x^3*y^3-x^3*y+x^2*y^4+x^2*y-x^2-x*y^4+x*y^3-x*y^2+x*y+y^3-2*y^2+y;
"The" (unique up to diamond+Mobius) three functions of degree 4 are:
x, y, and z := y*(x^2-x*y+y-1)/( (y-1)^2*x );
These functions x,y,z are modular units; their roots and poles are rational cusps.
The roots of x-1 and y-1 are rational cusps as well.
L = Q(x,y) = Q(x,z) has genus 5
The equation for x,z is:
Fxz := x^3*z^4-x^4*z^2-x^3*z^3-x^2*z^4+3*x^3*z^2+2*x^3*z+x*z^3-4*x^2*z-2*x*z^2-x^2+x+z;
Q(y,z) is a subfield of index 2 and genus 1
The equation for y,z is:
Fyz := y^2*z-y*z^2-y*z+z^2-y;
Gal(x) = S4, Gal(y) = D4, and Gal(z) = D4.
[note: this does not change if you replace Q by C]
So for all but finitely many rational numbers q,
the root of x-q resp. y-q resp z-q will give a pair (E,P) defined
over a number field K, with P of order 17,
and [K:Q] = 4, and Gal(K) is S4 resp. D4 resp. D4
where Gal(K) denotes Gal( normal closure of K ).
What are the exceptions?
Cuspidal-exceptions:
We are not interested in x in {0,1,infinity}, y in {0,1,infinity} or z in {0,infinity}
since those give only rational cusps on X1(17).
Any non-cuspidal exception will still give (E,P) over K with [K:Q]=4 because
there are no non-cuspidal points over fields of degrees 1,2,3.
So for D4, a non-cuspidal exception should have Gal(K) = C4 or C2 x C2
since these are the only transitive subgroups.
For S4, a non-cuspidal exception should have Gal(K) a subset of D4 or A4
(the maximal subgroups of S4 are D4, A4, or S3 but the last one is not transitive
so we can omit it).
A rational point on one of the following three curves means that G would be a subgroup of A4
(recall: not interested in cuspidal cases x in {0,1,infinity}, y in {0,1,infinity} or z in {0,infinity})
Px := T^2 - RemoveSquares( discrim(Fxy,y) );
Py := T^2 - RemoveSquares( discrim(Fxy,x) );
Pz := T^2 - RemoveSquares( discrim(Fxz,x) );
Px := T^2-x*(4*x^12-23*x^11+58*x^10-95*x^9+82*x^8-124*x^7+136*x^6-17*x^5-34*x^4-45*x^3+30*x^2+5*x-4);
Py := T^2-y^8+4*y^7-8*y^6+10*y^5-10*y^4+8*y^3-y^2-2*y-1;
Pz := T^2-z^8+2*z^7-7*z^6+4*z^5-13*z^4-4*z^3-7*z^2-2*z-1;
[note: Gal(y) = Gal(z) = D4, so a non-cuspidal rational point on Py or Pz gives a
point on X1(17) defined over a number field with group C2 x C2 since that is
the only transitive subgroup of D4 inside A4].
===================
The extension L/Q(x) has degree 4 and group S4. We computed the resolvent
polynomial for the subgroup D4 and found this:
Pxx := T^3+(-3*x^6+6*x^4-2*x^3-11*x^2+10*x-3)*T^2
+(3*x^12-16*x^11+36*x^10-44*x^9+98*x^8-204*x^7+198*x^6-68*x^5+15*x^4-48*x^3+50*x^2-20*x+3)*T
-(x^3-2*x^2+3*x-1)^2*(x^6-6*x^5+6*x^4-x^2-2*x+1)^2 ;
which is birational to:
Pxx := x*T^3-x*(x-1)^3*T^2+(x-1)*(3*x^2-1)*T+x^2*(x-1);
===================
The extension L/Q(y) has degree 4 and group D4. The polynomial Py tested
for C2 x C2 = D4 intersect A4.
Remains to test for the subgroup C4. For this, we need a resolvent polynomial.
Denote the 4 roots of Fxy (viewed as univariate polynomial in x over the field Q(y))
as R1,R2,R3,R4 ordered in such a way that Q(y)(R1) = Q(y)(R3).
So a potential C4 group would permute them as R1 -> R2 -> R3 -> R4 -> R1 and would
thus leave this invariant:
C4inv := R1 * R2^2 + R2 * R3^2 + R3 * R4^2 + R4 * R1^2;
That means that C4inv satisfies a degree 2 polynomial in Q(y)[T] and any non-cuspidal
rational point on that curve will give us C4.
Simplify that polynomial as: T^2 - RemoveSquares( discriminant ) and we get:
Pyy := T^2-(y^8-4*y^7+8*y^6-10*y^5+10*y^4-8*y^3+y^2+2*y+1)*(y-1)*y*(y^2-y-4) ;
===================
The extension L/Q(z) has degree 4 and group D4. Again, compute the minpoly of C4inv,
simplify, and we get
Pzz := T^2-(z^4-2*z^3+3*z^2+2*z+1)*(z^8-2*z^7+7*z^6-4*z^5+13*z^4+4*z^3+7*z^2+2*z+1);
===================
Next, we did a simple search for non-cuspidal points on Px, Py, Pz, Pxx, Pyy, Pzz.
The only one we found was y = 1/2 on Pyy
giving rise to a non-cuspidal point on X1(17) defined over a number field with group C4.
Are there any other examples, or is that it?