Elliptic curves over padic fields

class sage.schemes.elliptic_curves.ell_padic_field.EllipticCurve_padic_field(K, ainvs)

Bases: sage.schemes.elliptic_curves.ell_field.EllipticCurve_field, sage.schemes.hyperelliptic_curves.hyperelliptic_padic_field.HyperellipticCurve_padic_field

Elliptic curve over a padic field.

EXAMPLES:

sage: Qp=pAdicField(17)
sage: E=EllipticCurve(Qp,[2,3]); E
Elliptic Curve defined by y^2  = x^3 + (2+O(17^20))*x + (3+O(17^20)) over 17-adic Field with capped relative precision 20
sage: E == loads(dumps(E))
True

frobenius(P=None)

Return the Frobenius as a function on the group of points of this elliptic curve.

EXAMPLES:

sage: Qp = pAdicField(13)
sage: E = EllipticCurve(Qp,[1,1])
sage: type(E.frobenius())
<... 'function'>
sage: point = E(0,1)
sage: E.frobenius(point)
(0 : 1 + O(13^20) : 1 + O(13^20))

Check that trac ticket #29709 is fixed:

sage: Qp = pAdicField(13)
sage: E = EllipticCurve(Qp,[0,0,1,0,1])
sage: E.frobenius(E(1,1))
Traceback (most recent call last):
...
NotImplementedError: Curve must be in weierstrass normal form.
sage: E = EllipticCurve(Qp,[0,1,0,0,1])
sage: E.frobenius(E(0,1))
(0 : 1 + O(13^20) : 1 + O(13^20))