Projection Maps#
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 \(N\).
- ProjectionMap(CN, N, CM, M): Crv, RngIntElt, Crv, RngIntElt -> MapSch#
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 \(CN \rightarrow CM\) that corresponds to \(z \mapsto z\) in terms of the upper half-plane quotient models and to \((E,C) \mapsto (E,(N/M)C)\) 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 \(N\).
- ProjectionMap(CN, N, CM, M, r): Crv, RngIntElt, Crv, RngIntElt, RngIntElt -> MapSch#
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 \(CN \rightarrow CM\) that corresponds to \(z \mapsto rz\) in terms of the upper half-plane quotient models and to
\[(E,C) \mapsto (E/(N/r)C,(N/(Mr))C/(N/r)C)\]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 \(N\).
- Example: Sm Mod Crvs Proj Ex (ex-3882df)#
We use a 3-projection (case \(r=3\) in the above) in our extended example at the end of the section. Here, we just give some simple examples of the calls.
> 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