Parametrized Structures

The modular curves X_0(N) parametrize elliptic curves together with the structure of a cyclic isogeny, or equivalently, a cyclic subgroup scheme of the N-torsion of the elliptic curve. Since the algorithm specifying the isogeny implements an elliptic surface over the modular curve, singularities of the chosen surface may obstruct the construction of the corresponding isogeny.

Isogeny(E,P) : CrvEll, Pt -> MapCrvEll

Given an elliptic curve E and a point P on some X_0(N) corresponding to a cyclic level structure on E, returns an isogeny f: E -> F corresponding to P. The isogeny is defined by the formulae of Velu such that the pull-back of the invariant differential of F is the invariant differential on E.

SubgroupScheme(E,P) : CrvEll, Pt -> CrvEllSubgroup

Given an elliptic curve E and a point P on some X_0(N) corresponding to a cyclic level structure on E, returns the subgroup scheme of E parameterized by X_0(N).

Example CrvMod_Parametrized subgroup schemes (H107E3)

> A2 := AffineSpace(RationalField(),2);
> X0 := ModularCurve(A2,"Canonical",7);
> K0<u,j> := FunctionField(X0);
> j;
(u^8 + 28*u^7 + 322*u^6 + 1904*u^5 + 5915*u^4 + 8624*u^3 + 4018*u^2 + 
   748*u + 49)/u

> E := EllipticCurveFromjInvariant(j);
> ModuliPoints(X0,E);
[ (u, (u^8 + 28*u^7 + 322*u^6 + 1904*u^5 + 5915*u^4 + 8624*u^3 + 
  4018*u^2 + 748*u + 49)/u) ]
> P := $1[1];
> SubgroupScheme(E,P);
Subgroup of E defined by x^3 + (-1/4*u^4 - 9/2*u^3 - 115/4*u^2 - 
    129/2*u - 49/4)/(u^4 + 14*u^3 + 63*u^2 + 70*u - 7)*x^2 + 
    (1/48*u^8 + 3/4*u^7 + 269/24*u^6 + 268/3*u^5 + 19315/48*u^4 + 
    2968/3*u^3 + 9095/8*u^2 + 4309/12*u + 539/16)/(u^8 + 28*u^7 + 
    322*u^6 + 1904*u^5 + 5915*u^4 + 8624*u^3 + 4018*u^2 - 980*u + 
    49)*x + (-1/1728*u^12 - 1/32*u^11 - 47/64*u^10 - 8593/864*u^9 - 
    24767/288*u^8 - 142429/288*u^7 - 1096207/576*u^6 - 1378861/288*u^5 - 
    2136671/288*u^4 - 5444417/864*u^3 - 1377511/576*u^2 - 116185/288*u
    - 43169/1728)/(u^12 + 42*u^11 + 777*u^10 + 8246*u^9 + 54810*u^8 + 
    233730*u^7 + 628425*u^6 + 999306*u^5 + 801738*u^4 + 159838*u^3 - 
    93639*u^2 + 10290*u - 343)

> X0 := ModularCurve(A2,"Atkin",7);
> K0<u,j> := FunctionField(X0);
> j;
j
> E := EllipticCurveFromjInvariant(j);
> P := X0![u,j];
> SubgroupScheme(E,P);
Subgroup of E defined by x^3 + (7/4*u^9 + 1757/4*u^8 + 
    7819*u^7 - 73353*u^6 - 4884663/2*u^5 - 21811713/2*u^4 + 
    105047903*u^3 - 2*u^2*j + 1015540783*u^2 - 316*u*j + 
    8498533055/4*u - 2906*j - 4282204275/4)/(u^9 - 497*u^8 - 20532*u^7
    - 81388*u^6 + 4746950*u^5 + 56167290*u^4 - 1279028*u^3 - 
    2650748588*u^2 - 11224313439*u - 8626202865)*x^2 + (11/16*u^13 
    + 16939/48*u^12 + 1284239/24*u^11 + 50935457/24*u^10 + 
    1264382699/48*u^9 - 12214042943/48*u^8 - 124732865851/12*u^7 - 
    2/3*u^6*j - 1166543120669/12*u^6 - 1120/3*u^5*j + 
    839150370599/48*u^5 - 149162/3*u^4*j + 327535918621885/48*u^4 - 
    5354480/3*u^3*j + 409332919681469/8*u^3 - 81934006/3*u^2*j + 
    3407781609403081/24*u^2 - 573898352/3*u*j + 507049332829455/16*u -
    506125770*j - 5872301570153475/16)/(u^13 - 989*u^12 + 201198*u^11 
    + 21056034*u^10 + 657230331*u^9 + 6165550233*u^8 - 88238124492*u^7
    - 2578695909108*u^6 - 20624257862361*u^5 + 1318238025445*u^4 + 
    1038081350842750*u^3 + 6551865190346034*u^2 + 15514646620480317*u 
    + 9981405213700095)*x + (881/12096*u^17 + 687419/12096*u^16 + 
    8205629/504*u^15 + 390913160/189*u^14 + 49705687373/432*u^13 + 
    452515233571/144*u^12 + 2547613976705/72*u^11 - 37/504*u^10*j - 
    70616178199583/252*u^10 - 14107/252*u^9*j - 
    9780643994119277/672*u^9 - 8407621/504*u^8*j - 
    1169550002628136235/6048*u^8 - 129828949/63*u^7*j - 
    1332327646113277991/1512*u^7 - 4054218179/36*u^6*j + 
    127346633518027259/18*u^6 - 59532587759/18*u^5*j + 
    57066137174776820885/432*u^5 - 2072116237295/36*u^4*j + 
    377501715837964912373/432*u^4 - 38734075973053/63*u^3*j + 
    1408654650164439479513/504*u^3 - 2001461342219281/504*u^2*j + 
    70839088386805558805/28*u^2 - 399602464019187/28*u*j - 
    3877734333962296889325/448*u - 1229361350588865/56*j - 
    8491275377159013948375/448)/(u^17 - 1481*u^16 + 664992*u^15 - 
    55683056*u^14 - 13999885052*u^13 - 854937578868*u^12 - 
    24503558413296*u^11 - 316596143942688*u^10 + 692774466143310*u^9 +
    87641547175946290*u^8 + 1328002620135144640*u^7 + 
    8491254793740345168*u^6 - 3391803340001770028*u^5 - 
    462587327970641785700*u^4 - 3479022140596534545648*u^3 - 
    12353559598789551006912*u^2 - 20875996991798942070951*u - 
    11549513916988021024785)