direct computation of .formal()[1] for elliptic-curve morphisms

[yyyyx4]

Follow-up to #33216: We can compute .formal()[1] directly, i.e., without going through .formal(). This is beneficial as the latter uses the rational maps, which can be very expensive.

• Composite isogenies have multiplicative .formal()[1].
• Isogenies using Vélu's or Kohel's formulas are normalized, hence .formal()[1] == 1. We only need to account for pre- and post-isomorphism.
• Weierstrass isomorphisms (u,r,s,t) have .formal()[1] == u.

Same example as for #33216:

E = EllipticCurve(j=GF(431^2)(4))
phi = E.isogeny(2^99*E.gens()[0])
_ = phi.dual()
%timeit phi._EllipticCurveIsogeny__dual = None; phi.dual()

Sage 9.6:

537 ms ± 6.75 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

This branch:

294 ms ± 1.71 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

CC: @defeo @JohnCremona @categorie

Component: elliptic curves

Author: Lorenz Panny

Branch/Commit: fb571d3

Reviewer: John Cremona

Issue created by migration from https://trac.sagemath.org/ticket/33865

[JohnCremona]

comment:2

This looks great, and well worth doing. My only comment is that all these _scaling_factor() methods should be user-acessible, i.e. not have the underscore, since this is a useful number associated to an isogeny -- one that I have often needed to access.

At the same time, perhaps we can have in the docstring a definition of this "scaling factor". I would say something like this: it is the factor u (in the base field) such that $\phi^*\omega_2 = u \omega_1$, where $\phi:E_1\to E_2$ is the isogeny and for $i=1,2$, $\omega_i$ is the standard Weierstrass differential, i.e. $dx/(2y+a_1x+a_3)$.

[Sorry, I forgot that trac does not understand latex.]

[sagetrac-git]

Changed commit from 2e88ff0 to fb571d3

[sagetrac-git]

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

fb571d3

turn .scaling_factor() into a public function

[yyyyx4]

comment:4

Thank you for the suggestion! I changed it to .scaling_factor() and added your definition (with minor tweaks) to the documentation.

[JohnCremona]

comment:5

Sorry about the delay in coming back to this.
I would give a positive review, but do not understand the "test fail" flag. I looked at the logs and tere were no failing tests, the log just stops in the middle.

Is it possible to get the test to re-run?

[yyyyx4]

comment:6

The patchbot log here on Trac is fully green and looks complete over here. Maybe your browser hadn't finished loading it all the way to the end?

On GitHub Actions (the source of the alarmingly red "Build & Test | failing" icon), there is a single test failure, but it seems entirely unrelated to this patch:

• https://github.com/sagemath/sagetrac-mirror/runs/6491511567#step:9:6564

sage -t --random-seed=255890665791265155015760495860109238381 sage/parallel/map_reduce.py
**********************************************************************
File "sage/parallel/map_reduce.py", line 1151, in sage.parallel.map_reduce.RESetMapReduce.start_workers
Failed example:
    all(not w.is_alive() for w in S._workers)
Expected:
    True
Got:
    False
**********************************************************************
1 item had failures:
   1 of  10 in sage.parallel.map_reduce.RESetMapReduce.start_workers
    [294 tests, 1 failure, 41.75 s]

[JohnCremona]

comment:7

Thanks for the quick help. I do now see the whole log file. I'll give this a positive review then.

[JohnCremona]

Reviewer: John Cremona

[yyyyx4]

comment:8

Thank you!

[vbraun]

Changed branch from public/elliptic_curve_morphisms_scaling_factor to fb571d3