The database contains information allowing the reconstruction of the standard projection maps between the models of the level N and level M curves for any M dividing N.
The curves CN and CM should be base changes to the same characteristic zero field K of the small modular database curves of levels N and M with M | N.Returns the natural projection map CN -> CM that corresponds to z |-> z in terms of the upper half-plane quotient models and to (E, C) |-> (E, (N/M)C) in the moduli space interpretation where non-cuspidal points correspond to isomorphism classes of an elliptic curves E with a cyclic subgroup C of order N.
The curves CN and CM should be base changes to the same characteristic zero field K of the small modular database curves of levels N and M with M | N and r should be a positive integer divisor of N/M.Returns the projection map CN -> CM that corresponds to z |-> rz in terms of the upper half-plane quotient models and to (E, C) |-> (E/(N/r)C, (N/(Mr))C/(N/r)C) in the moduli space interpretation where non-cuspidal points correspond to isomorphism classes of an elliptic curves E with a cyclic subgroup C of order N.
> C63<x,y,z> := SmallModularCurve(63);
> C63;
Curve over Rational Field defined by
x^5*y - 2*x^4*y^2 + 3*x^3*y^3 - 2*x^2*y^4 + x*y^5 - 2*x^3*z^3 + x^2*y*z^3 +
x*y^2*z^3 - 2*y^3*z^3 + z^6
> C21 := SmallModularCurve(21);
> C21;
Elliptic Curve defined by y^2 + x*y = x^3 - 4*x - 1 over Rational Field
> C3 := SmallModularCurve(3);
> C3;
Curve over Rational Field defined by
0
> ProjectionMap(C63,63,C21,21);
Mapping from: Crv: C63 to CrvEll: C21
with equations :
8*x^3*y - 6*x^2*y^2 + 6*x*y^3 + 2*y^4 - 7*x^3*z + 15*x^2*y*z - 10*x*y^2*z +
4*y^3*z - 7*x*y*z^2 + 7*y^2*z^2 + 5*x*z^3 - 15*y*z^3 - 2*z^4
-7*x^4 + 17*x^3*y - 18*x^2*y^2 + 11*x*y^3 - y^4 + 3*x^2*y*z + 5*x*y^2*z +
5*y^3*z + 21*x^2*z^2 - 28*x*y*z^2 + 7*y^2*z^2 + x*z^3 - 3*y*z^3 - 13*z^4
-4*x^3*y + 3*x^2*y^2 - 3*x*y^3 - y^4 + 3*x^2*y*z - 2*x*y^2*z - 2*y^3*z +
7*x*y*z^2 - 7*y^2*z^2 + x*z^3 + 4*y*z^3 + z^4
> ProjectionMap(C63,63,C3,3,7); //7-projection
Mapping from: Crv: C63 to Crv: C3
Composition of Mapping from: Crv: C63 to Crv: C
with equations :
x^2 - x*y + y^2 + 2*x*z - y*z - 2*z^2
x^2 - x*y + y^2 - x*z + 2*y*z - 2*z^2
x^2 - x*y + y^2 - x*z - y*z + z^2 and
Mapping from: Crv: C63 to Curve over Rational Field defined by
0
with equations :
x^3 + 3*x^2*y - 3*x*y^2 + 4*y^3 - 3*x^2*z - 2*z^3
x^2*z and
Mapping from: Curve over Rational Field defined by
0 to Curve over Rational Field defined by
0
with equations :
27*$.2
$.1 and
Mapping from: Curve over Rational Field defined by
0 to Crv: C3
with equations :
$.1^3
$.1^2*$.2 + 9*$.1*$.2^2 + 27*$.2^3