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### n-torsion subgroups on Elliptic Curves defined on some field

if E an elliptic is defined on K, then E[n] is a subgroup of an Elliptic curve defined on an algebraic closure of K, its the set of points P of this curve whose order is dividing n

G = E.torsion_subgroup() gives the torsion subgroup in E(K), but is there a method giving the E[n] ?

thanks

### n-torsion subgroups on Elliptic Curves defined on some field

if E an elliptic is defined on K, then E[n] is a subgroup of an Elliptic curve defined on an algebraic closure of K, its the set of points P of this curve whose order is dividing n

G = E.torsion_subgroup() gives the torsion subgroup in E(K), but is there a method giving the E[n] ?

For example, for the Elliptic Curve y^2=x^3-2

if we denote by alpha the cubic root of 2 and J a complex cubic root of unity, and O the point at infinity

E[2] = {O,(alpha,0), (alphaJ,0),(alphaJ^2,0)}

thanks

### n-torsion subgroups on Elliptic Curves defined on some field

if E an elliptic is defined on K, then E[n] is a subgroup of an Elliptic curve defined on an algebraic closure of K, its the set of points P of this curve whose order is dividing n

G = E.torsion_subgroup() gives the torsion subgroup in E(K), but is there a method giving the E[n] ?

For example, for the Elliptic Curve y^2=x^3-2

if we denote by alpha the cubic root of 2 and J a complex cubic root of unity, and O the point at infinity

E[2] = {O,(alpha,0), (alphaJ,0),(alpha(alpha * J,0),(alpha * J^2,0)}

thanks

### n-torsion subgroups on Elliptic Curves defined on some field

if E an elliptic is defined on K, then E[n] is a subgroup of an Elliptic curve defined on an algebraic closure of K, its the set of points P of this curve whose order is dividing n

G = E.torsion_subgroup() gives the torsion subgroup in E(K), but is there a method giving the E[n] ?

For example, for the Elliptic Curve y^2=x^3-2

if we denote by alpha the cubic root of 2 and J a complex cubic root of unity, and O the point at infinity

E[2] = {O,(alpha,0), (alpha * J,0),(alpha * J^2,0)}

I tried the following

E = EllipticCurve(CC,[0,-2]) O = E(0)

O.division_points(2)

and got [pretty close but not the right result, and not in an algebraic form anyway]

 [(0.000000000000000 : 1.00000000000000 : 0.000000000000000),
(1.25992104989487 : 0.000000000000000 : 1.00000000000000)]


thanks

### n-torsion subgroups on Elliptic Curves defined on some field

if E an elliptic is defined on K, then E[n] is a subgroup of an Elliptic curve defined on an algebraic closure of K, its the set of points P of this curve whose order is dividing n

G = E.torsion_subgroup() gives the torsion subgroup in E(K), but is there a method giving the E[n] ?

For example, for the Elliptic Curve y^2=x^3-2

if we denote by alpha the cubic root of 2 and J a complex cubic root of unity, and O the point at infinity

E[2] = {O,(alpha,0), (alpha * J,0),(alpha * J^2,0)}

I tried the following

E = EllipticCurve(CC,[0,-2])

O = E(0)

O.division_points(2)

and got [pretty close but not the right result, and not in an algebraic form anyway]

 [(0.000000000000000 : 1.00000000000000 : 0.000000000000000),
(1.25992104989487 : 0.000000000000000 : 1.00000000000000)]


thanks

### n-torsion subgroups on Elliptic Curves defined on some field

if E an elliptic is defined on K, then E[n] is a subgroup of an Elliptic curve defined on an algebraic closure of K, its the set of points P of this curve whose order is dividing n

G = E.torsion_subgroup() gives the torsion subgroup in E(K), but is there a method giving the E[n] ?

For example, for the Elliptic Curve y^2=x^3-2y^2=x^3-2 defined on Q

if we denote by alpha the cubic root of 2 and J a complex cubic root of unity, and O the point at infinity

E[2] = {O,(alpha,0), (alpha * J,0),(alpha * J^2,0)}

I tried the following

E = EllipticCurve(CC,[0,-2])

O = E(0)

O.division_points(2)

and got [pretty close but not the right result, and not in an algebraic form anyway]

 [(0.000000000000000 : 1.00000000000000 : 0.000000000000000),
(1.25992104989487 : 0.000000000000000 : 1.00000000000000)]


thanks