mutative algebra – Pleasant forms for the Weierstrass equation

The motivation for this question comes from the study of elliptic curves, but the question itself is about the choice of convenient algebraic transformations.

Fix a base ring (commutative, with a multiplicative identity) $ R $. We consider the polynomial equation
y ^ 2 + a_1xy + a_3y = x ^ 3 + a_2x ^ 2 + a_4x + a_6

where $ a_i $ stay in bed $ R $. The allowed coordinate transformations are
x = x & # 39; + r, qquad y = y & # 39; + sx & # 39; + t, qquad r, s, t in R.

It can be verified that under such a transformation, the shape of the polynomial equation is preserved but $ a_i $ change its value (in some explicit way I don't want to write at this time).

Yes, both $ 2 $ Y $ 3 $ are units in $ R $, there is a unique choice of coordinates so that $ a_1 = a_2 = a_3 = 0 $.

Yes $ 2 = 0 in R $ (resp. $ 3 = 0 in R $) there are sub-boxes, but you can choose (not exclusively) a good form for the equation and describe all the transformations that fix it (Deligne, Courbes elliptique: formulaire).

I want to consider $ R = mathbb {Z} / m mathbb {Z} $ for $ m> $ 3 (particularly $ m = $ a power of $ 2 $ or $ 3 $) Has anyone discovered what is a good way for our equation on this base ring (and what are the transformations that preserve it)?

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Apex Glide Team: solutions ready for productive work and a pleasant stay



  • Telegram: @apexglideteam
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  • Skype: live: 5f6e68fde6610713
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Human interface guide: look for options to make the user interface more pleasant and understandable

We are building a lot of small strategic tools. Most of them are really simple and should be very easy to understand.

One of them is the construction to execute the comparison of several elements. The comparison is made in pairs.

It sounds easy, but the best user interface that I can think of is the following:
enter the description of the image here

Here, the user needs to UPDATE one of the options in the pair, then the process changes to the next, and so on.

This user interface feels really strange, especially in extreme cases when the content of the elements becomes very long or very short (as in the images above).
It just is not sexy absolutely.

I was hoping to get some kind of guideline or suggestion to guide me in the right direction.

Thank you,