EllipticCurve.period_lattice() should automatically use the canonical embedding for embedded number fields?

[user202729]

Problem Description

Currently, period_lattice() requires passing an embedding argument for number field except for ℚ even though it is already an embedded number field.

sage: K.<ω> = number_field_elements_from_algebraics([QQbar((-1)^(2/3))], embedded=True, minimal=True)[0]
sage: K
Number Field in zeta6 with defining polynomial x^2 - x + 1 with zeta6 = 0.500000000000000? + 0.866025403784439?*I
sage: E = EllipticCurve(K, [1, 0])
sage: E.period_lattice()
Traceback (most recent call last)
...
TypeError: EllipticCurve_number_field.period_lattice() missing 1 required positional argument: 'embedding'

Proposed Solution

We make it automatically do the equivalent of the following.

sage: E.period_lattice(E.base_field().coerce_embedding())
Period lattice associated to Elliptic Curve defined by y^2 = x^3 + x over Number Field in zeta6 with defining polynomial x^2 - x + 1 with zeta6 = 0.500000000000000? + 0.866025403784439?*I with respect to the embedding Generic morphism:
  From: Number Field in zeta6 with defining polynomial x^2 - x + 1 with zeta6 = 0.500000000000000? + 0.866025403784439?*I
  To:   Algebraic Field
  Defn: zeta6 -> 0.500000000000000? + 0.866025403784439?*I

---

There's a caveat though, some special fields have coerce_embedding always defined.

sage: K.<ω> = number_field_elements_from_algebraics([QQbar((-1)^(2/3))], embedded=False, minimal=True)[0]
sage: K
Cyclotomic Field of order 6 and degree 2
sage: K.coerce_embedding()
Generic morphism:
  From: Cyclotomic Field of order 6 and degree 2
  To:   Complex Lazy Field
  Defn: zeta6 -> 0.500000000000000? + 0.866025403784439?*I

Maybe we use an isinstance() check or something? (alternatively we modify coerce_embedding() of cyclotomic field to not automatically convert unless an embedding is explicitly specified by the user)

Alternatives Considered

Or we can just raise an error in period_lattice but suggest the user they could use the second form. It would make coerce_embedding() more discoverable.

Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.