Improve downloaded search results for elliptic curves over number fields

[rvisser7]

When downloading the search results for a set of elliptic curves over number fields $\mathbb{Q}(\alpha)$ (for any language Sage, Magma, Pari, or Oscar) , the data unfortunately doesn't include an easy way to actually use the elliptic curves as objects in the respective language. Currently, it only contains a string of the latexed Weierstrass equation, which isn't particularly useful when trying to run your own computations on the data.

Example: https://beta.lmfdb.org/EllipticCurve/?field=2.0.111.1&conductor_norm=1
When downloading to Sage all elliptic curves with conductor norm 1 over the base field 2.0.111.1, the file contains the following:

columns = ["short_label", "short_class_label", "field_label", "conductor_norm", "rank", "torsion_structure", "cm", "sato_tate_group", "equation"]
data = [
["1.1-a1", "1.1-a", "2.0.111.1", 1, 1, [], 0, "1.2.A.1.1a", "{y}^2+{y}={x}^3+\\left(-a+1\\right){x}^2+\\left(-3a-16\\right){x}+7a-31"],
["1.1-a2", "1.1-a", "2.0.111.1", 1, 1, [], 0, "1.2.A.1.1a", "{y}^2+{y}={x}^3+\\left(-a+1\\right){x}^2+\\left(-563a-1486\\right){x}-14287a+263"],
["1.1-b1", "1.1-b", "2.0.111.1", 1, 1, [], 0, "1.2.A.1.1a", "{y}^2+{y}={x}^3+a{x}^2+\\left(3a-19\\right){x}-7a-24"],
["1.1-b2", "1.1-b", "2.0.111.1", 1, 1, [], 0, "1.2.A.1.1a", "{y}^2+{y}={x}^3+a{x}^2+\\left(563a-2049\\right){x}+14287a-14024"]
]

def create_record(row):
    out = {col: val for col, val in zip(columns, row)}
    return out

I think it would definitely be a lot more helpful if the downloaded search results could create an EllipticCurve object, as is currently done for elliptic curves over Q! E.g. It would be great to have something similar to what is currently implemented for downloading elliptic curves over Q to Sage:

columns = ["lmfdb_label", "lmfdb_iso", "class_size", "conductor", "rank", "torsion_structure", "cm_discriminant", "ainvs"]
data = [
["32.a1", "32.a", 4, 32, 0, [2], -16, [0, 0, 0, -11, -14]],
["64.a1", "64.a", 4, 64, 0, [2], -16, [0, 0, 0, -44, -112]]
]

def create_record(row):
    out = {col: val for col, val in zip(columns, row)}
    curve = EllipticCurve(out["ainvs"])
    out["curve"] = curve
    return out

[roed314]

I agree, this would be a good improvement.

[rvisser7]

Great, I'd be happy to have a go at improving this in the new year :).

I see there is also a related (still open) issue #4810, although perhaps that one was more focused on improving the search result downloads for elliptic curves over Q?