The torsion or division polynomials are the polynomials defining the subschemes of n-torsion points on the elliptic curve.
Returns the homogeneous defining polynomial for the elliptic curve E.
Given an elliptic curve E and an integer n, returns the n-th division polynomial as a univariate polynomial over the base ring of E; its roots are the x-coordinates of the nonzero n-torsion points, each occurring once. The second return value is this polynomial divided by the univariate 2-torsion polynomial if n is even. The third argument is the cofactor, equal to the univariate 2-torsion polynomial if n is even, and 1 otherwise.If a polynomial is passed as a third argument then the division polynomial is computed efficiently modulo that polynomial.
Returns the multivariate 2-torsion polynomial 2y + a1x + a3 of the elliptic curve E, as a bivariate polynomial.
> K := GF(101); > E := EllipticCurve([ K | 1, 1]); > Roots(DivisionPolynomial(E, 5)); [ <86, 1>, <46, 1> ] > [ P : P in RationalPoints(E) | 5*P eq E!0 ]; [ (86 : 34 : 1), (0 : 1 : 0), (46 : 25 : 1), (86 : 67 : 1), (46 : 76 : 1) ]It is worth noting that even if the roots of the division polynomial lie in the field, the corresponding points may not, lying instead in a quadratic extension.
> Roots(DivisionPolynomial(E, 9)); [ <4, 1>, <17, 1>, <28, 1>, <34, 1>, <77, 1> ] > Points(E, 4); [] > K2<w> := ext<K | 2>; > Points(E(K2), 4); [ (4 : 82*w + 57 : 1), (4 : 19*w + 44 : 1) ] > Order($1[1]); 9
Let E be an elliptic curve with complex multiplication by an order O in an imaginary quadratic field K, and let α ∈O. The endomorphism [α] : E to E acts on x-coordinates by a rational function x([α]P) = φα(x) / ψα(x)e, generalising the classical division polynomials; here e = 2 when N(α) is odd and e = 1 when the kernel is entirely 2-torsion (a prime above 2), matching the convention of MultiplicationByMMap; for other even N(α), where the kernel mixes 2-torsion with higher-order points, no single e works and CMEndomorphism should be used for the map instead. The endomorphism is normalised so that [α] * ω = α ω on the invariant differential ω; equivalently φα has leading coefficient 1/α2. The element α may be prime, a prime power, or composite: a composite α is handled by composing the prime endomorphisms, which requires K to have class number 1. Currently E must be defined over Q or over a field containing the CM field K (when K has class number greater than 1 the CM endomorphisms are defined over the ring class field, so E must be given there), and positive characteristic (finite fields) is not supported.
Given an elliptic curve E with complex multiplication by an order containing α, returns the complex multiplication division polynomial ψα: the monic squarefree polynomial over the CM field K whose roots are the x-coordinates of the nonzero points in the kernel of the endomorphism [α] (each occurring once). For a prime α lying above an odd rational prime p its degree is (p2 - 1)/2 when p is inert and (p - 1)/2 when p splits or ramifies. For a prime above 2 the degree is 3 when 2 is inert (the kernel is all of E[2]) and 1 when 2 splits or ramifies. For a split prime the two prime ideals above p give distinct kernels, and the polynomial returned is the one belonging to the specific generator α, its conjugate giving ψbarα.
Given an elliptic curve E with complex multiplication and an ideal a of its CM order, returns the a-division polynomial: the monic polynomial whose roots are the x-coordinates of the nonzero points of the a-torsion E[a] (the points killed by every element of a). For a principal ideal a = (α) this equals DivisionPolynomial(E, alpha) and depends only on a, not on the choice of generator. Non-principal ideals (which occur only when the CM field has class number greater than 1) are not currently supported.
Given an elliptic curve E with complex multiplication by an order containing α, returns the numerator φα of the multiplication-by-α map, so that x([α]P) = φα(x) / ψα(x)e where ψα is DivisionPolynomial(E, alpha) and e = 2 for N(α) odd, e = 1 when the kernel is entirely 2-torsion (a prime above 2). For other even N(α), where the kernel mixes 2-torsion with higher-order points, no single e reproduces x([α]P) from ψα; use CMEndomorphism for the map in that case. This is the complex multiplication analogue of the multiplication map MultiplicationByMMap(E, n) for a rational integer n. The polynomial φα has degree N(α) and leading coefficient 1/α2.
Given an elliptic curve E with complex multiplication by an order containing α, returns the CM endomorphism [α] : E to E as a map, normalised so that [α] * ω = α ω on the invariant differential. On x-coordinates it acts by the rational function of MultiplicationPolynomial and DivisionPolynomial above. This is the complex multiplication analogue of MultiplicationByMMap(E, n) for a rational integer n.
> K<w> := QuadraticField(-3); > O<a> := MaximalOrder(K); > E := EllipticCurve([K | 0, 0, 0, 0, -1]); > _, D := HasComplexMultiplication(E); > D; -3 > alpha := 2 + a; > Norm(alpha); 7 > psi := DivisionPolynomial(E, alpha); > psi; x^3 + 1/7*(-6*w + 2) > ChangeRing(DivisionPolynomial(E, 7), K) mod psi eq 0; true > phi := MultiplicationPolynomial(E, alpha); > phi; 1/98*(-5*w + 11)*x^7 + 1/7*(26*w + 10)*x^4 + 1/7*(-16*w - 32)*x