Subsections

• Creation of an Elliptic Curve
• Creation of Points
• Invariants

EllipticCurve([a1,a2,a3,a4,a6]) : [RngElt] -> GeomEC

Given a sequence of elements of a ring R, create the elliptic curve over R defined by taking the elements of the sequence as Weierstrass coefficients. The length of the sequence must either be 2, that is s=[a, b], in which case E is defined by y^2z = x^3 + axz^2 + bz^3, or 5, that is s=[a_1, a_2, a_3, a_4, a_6], in which case E is given by y^2z + a_1xyz + a_3yz^2=x^3 + a_2x^2z + a_4xz^2 + a_6z^3. Currently R must be field; if integers are given, they will be coerced into Q.

E ! [x, y] : GeomEC, [RngElt] -> GeomECElt

elt< E | x, y, z > : GeomEC, RngElt, RngElt, RngElt -> GeomECElt

elt< E | x, y > : GeomEC, RngElt, RngElt -> GeomECElt

Given an elliptic curve E over R and coefficients x, y, z in R satisfying the equation for E, return the point P=(x, y, z) in E[R]. If z is not specified it is assumed to be 1.

Id(E) : GeomEC -> GeomECElt

E ! 0 : GeomEC, RngIntElt -> GeomECElt

Return the identity point (0:1:0) on the curve E.

Given an elliptic curve E, this function returns a sequence consisting of the Weierstrass coefficients of E; this is the sequence [a_1, a_2, a_3, a_4, a_6] such that E is defined by y^2z + a_1xyz + a_3yz^2=x^3 + a_2x^2z + a_4xz^2 + a_6z^3, unless a_1=a_2=a_3=0, in which case the sequence [a, b] is returned, and E is defined by y^2z = x^3 + axz^2 + bz^3.

This function returns a sequence of length 4 containing the b-invariants of the elliptic curve E, namely [b_2, b_4, b_6, b_8]. In terms of the coefficients a_1, a_2, a_3, a_4, a_6 these invariants are defined by

b_2 = a_1^2 + 4a_2
b_4 = a_1a_3 + 2a_4
b_6 = a_3^2 + 4a_6
b_8 = a_1^2a_6 + 4a_2a_6 - a_1a_3a_4 + a_2a_3^2 - a_4^2.

The common parent of these elements will be the field over which E is defined.

This function returns a sequence of length 2 containing the c-invariants of the elliptic curve E, namely [c_4, c_6]. In terms of the b-invariants b_2, b_4, b_6, b_8 these invariants are defined by

c_4 = b_2^2 - 24b_4
c_6 = b_2^3 + 36b_2b_4 - 216b_6.

Return the discriminant of the elliptic curve E, which equals -b_2^2b_8 - 8b_4^3 - 27b_6^2 + 9b_2b_4b_6, in terms of the b-invariants of the curve. The discriminant will be an element of the field of definition of E.

Returns the j-invariant of the elliptic curve E, which equals c_4^3/Delta, in terms of the c-invariants and the discriminant of the curve.

Given an elliptic curve E with integral coefficients defined over Q, return the sequence of primes dividing the minimal discriminant of E. These are the primes at which the minimal model for E has bad reduction; note that there may be other primes dividing the discriminant for the given model of E.

The conductor of an elliptic curve with integral coefficients defined over Q.

Given an elliptic curve E, defined over Q with integral coefficients, and a prime number p, this function returns the local Tamgawa number of E at p, which is the index in E[Q_p] of the subgroup E^0[Q_p] consisting of points with non-singular reduction modulo p.

Given an elliptic curve E, defined over Q with integral coefficients, this function returns the sequence of Tamagawa numbers at each of the bad primes of E, as defined above.

Given an elliptic curve E defined over Q with integral coeffients, as well as a prime number p this function returns the local information at the prime p as a 5-tuple, consisting of p, its multiplicity in the discriminant, its multiplicity in the conductor, the Tamagawa number at p and the Kodaira symbol.

LocalInformation(E) : GeomEC, RgIntElt -> [ Tup ]

Given an elliptic curve E this function returns a sequence of tuples, each of which contains the local information at a bad prime. The tuples consist of a bad prime p, ts multiplicity in the discriminant, its multiplicity in the conductor, the Tamagawa number at p and the Kodaira symbol.