more efficient method for number of real components of an elliptic curve over Q

[JohnCremona]

Define an elliptic curve, find its discriminant
and its number of real components:

sage: E = EllipticCurve([0, -1, 1, 0, 0])
sage: E.discriminant()
-11
sage: E.real_components()
1

The number of real components is 1 or 2 when the discriminant is negative or positive respectively. The current code does a lot of unnecessary work:

        invs = self.short_weierstrass_model().ainvs()
        x = rings.polygen(self.base_ring())
        f = x**3 + invs[3]*x + invs[4]
        if f.discriminant() > 0:
            return 2
        else:
            return 1

It is unnecessary to compute a short Weierstrass model, take its coefficients, construct a polynomial, and compute its discriminant, since E.discriminant() has the same sign as the discriminant computed here.

As well as fixing this we add a real_components() method for elliptic curves over number fields, which takes as a parameter a real embedding of the base field.

CC: @slel

Component: elliptic curves

Author: John Cremona

Branch/Commit: 41b446c

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/31433

[JohnCremona]

Commit: 2f0b0c1

[JohnCremona]

New commits:

f3c6f63	#31433: simplify real_components method for elliptic curves over QQ
2f0b0c1	#31433: add real_components method for elliptic curves over number fields

[JohnCremona]

Branch: u/cremona/31433

[fchapoton]

comment:4

two many final dots here:

+        Return the number of real components with respect to a real embedding of the base field..

[sagetrac-git]

Changed commit from 2f0b0c1 to 41b446c

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

41b446c

#31433: delete extra dot

[fchapoton]

Reviewer: Frédéric Chapoton

[fchapoton]

comment:6

ok, looks good. Avanti !

[JohnCremona]

comment:7

Replying to @fchapoton:

ok, looks good. Avanti !

Merci!

[vbraun]

Changed branch from u/cremona/31433 to 41b446c