Basics

Creation

# elliptic_curve — Function.

elliptic_curve([K::Field], x::Vector; check::Bool = true) -> EllipticCurve

Construct an elliptic curve with Weierstrass equation specified by the coefficients in x, which must have either length 2 or 5.

Per default, it is checked whether the discriminant is non-zero. This can be disabled by setting check = false.

Examples

julia> elliptic_curve(QQ, [1, 2, 3, 4, 5])
Elliptic curve with equation
y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5

julia> elliptic_curve(GF(3), [1, 1])
Elliptic curve with equation
y^2 = x^3 + x + 1

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# elliptic_curve_from_j_invariant — Function.

elliptic_curve_from_j_invariant(j::FieldElem) -> EllipticCurve

Return an elliptic curve with the given $j$-invariant.

Examples

julia> K = GF(3)
Prime field of characteristic 3

julia> elliptic_curve_from_j_invariant(K(2))
Elliptic curve with equation
y^2 + x*y = x^3 + 1

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Basic properties

# base_field — Method.

base_field(E::EllipticCurve) -> Field

Return the base field over which E is defined.

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base_field(C::HypellCrv) -> Field

Return the base field over which C is defined.

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# base_change — Method.

base_change(K::Field, E::EllipticCurve) -> EllipticCurve

Return the base change of the elliptic curve $E$ over $K$ if coercion is possible.

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# base_change — Method.

base_change(f, E::EllipticCurve) -> EllipticCurve

Return the base change of the elliptic curve $E$ using the map $f$.

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# coefficients — Method.

coefficients(E::EllipticCurve{T}) -> Tuple{T, T, T, T, T}

Return the Weierstrass coefficients of $E$ as a tuple (a1, a2, a3, a4, a6) such that $E$ is given by y^2 + a1xy + a3y = x^3 + a2x^2 + a4x + a6.

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# a_invariants — Method.

a_invariants(E::EllipticCurve{T}) -> Tuple{T, T, T, T, T}

Return the Weierstrass coefficients of $E$ as a tuple $(a_1,a_2,a_3,a_4,a_6)$ such that $E$ is given by $y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6$.

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# b_invariants — Method.

b_invariants(E::EllipticCurve{T}) -> Tuple{T, T, T, T}

Return the b-invariants of $E$ as a tuple $(b_2,b_4,b_6,b_8)$.

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# c_invariants — Method.

c_invariants(E::EllipticCurve{T}) -> Tuple{T, T}

Return the c-invariants of $E as a tuple $(c_4,c_6)$.

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# discriminant — Method.

discriminant(E::EllipticCurve) -> FieldElem

Return the discriminant of $E$.

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discriminant(C::HypellCrv{T}) -> T

Compute the discriminant of $C$.

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discriminant(O::AlgssRelOrd)

Returns the discriminant of $O$.

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# j_invariant — Method.

j_invariant(E::EllipticCurve) -> FieldElem

Compute the j-invariant of $E$.

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# equation — Method.

equation([R::MPolyRing,] E::EllipticCurve) -> MPolyRingElem

Return the equation defining the elliptic curve $E$ as a bivariate polynomial. If the polynomial ring $R$ is specified, it must by a bivariate polynomial ring.

Examples

julia> E = elliptic_curve(QQ, [1, 2, 3, 4, 5]);

julia> equation(E)
-x^3 - 2*x^2 + x*y - 4*x + y^2 + 3*y - 5

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# hyperelliptic_polynomials — Method.

hyperelliptic_polynomials([R::PolyRing,] E::EllipticCurve) -> PolyRingElem, PolyRingElem

Return univariate polynomials $f,h$ such that $E$ is given by $y^2+h\ast y=f$.

Examples

julia> E = elliptic_curve(QQ, [1, 2, 3, 4, 5]);

julia> hyperelliptic_polynomials(E)
(x^3 + 2*x^2 + 4*x + 5, x + 3)

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Points

    (E::EllipticCurve)(coords::Vector; check::Bool = true)

Return the point $P$ of $E$ with coordinates specified by coords, which can be either affine coordinates (length(coords) == 2) or projective coordinates (length(coords) == 3).

Per default, it is checked whether the point lies on $E$. This can be disabled by setting check = false.

Examples

julia> E = elliptic_curve(QQ, [1, 2]);

julia> E([1, -2])
Point  (1 : -2 : 1)  of Elliptic curve with equation
y^2 = x^3 + x + 2

julia> E([2, -4, 2])
Point  (1 : -2 : 1)  of Elliptic curve with equation
y^2 = x^3 + x + 2

# infinity — Method.

infinity(E::EllipticCurve) -> EllipticCurvePoint

Return the point at infinity with project coordinates $[0:1:0]$.

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# parent — Method.

parent(P::EllipticCurvePoint) -> EllipticCurve

Return the elliptic curve on which $P$ lies.

Examples

julia> E = elliptic_curve(QQ, [1, 2]);

julia> P = E([1, -2]);

julia> E == parent(P)
true

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# is_on_curve — Method.

is_on_curve(E::EllipticCurve, coords::Vector) -> Bool

Return true if coords defines a point on $E$ and false otherwise. The array coords must have length 2.

Examples

julia> E = elliptic_curve(QQ, [1, 2]);

julia> is_on_curve(E, [1, -2])
true

julia> is_on_curve(E, [1, -1])
false

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# + — Method.

+(P::EllipticCurvePoint, Q::EllipticCurvePoint) -> EllipticCurvePoint

Add two points on an elliptic curve.

Examples

julia> E = elliptic_curve(QQ, [1, 2]);

julia> P = E([1, -2]);

julia> P + P
Point  (-1 : 0 : 1)  of Elliptic curve with equation
y^2 = x^3 + x + 2

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# division_points — Method.

division_points(P::EllipticCurvePoint, m::Int) -> EllipticCurvePoint

Compute the set of points $Q$ defined over the base field such that $mQ=P$. Returns the empty list if no such points exist.

Examples

julia> E = elliptic_curve(QQ, [1, 2]);

julia> division_points(infinity(E), 2)
2-element Vector{EllipticCurvePoint{QQFieldElem}}:
 Point  (0 : 1 : 0)  of Elliptic curve with equation
y^2 = x^3 + x + 2
 Point  (-1 : 0 : 1)  of Elliptic curve with equation
y^2 = x^3 + x + 2

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