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Fraction must have unit denominator when studying Elliptic Curve over complex field? [closed]

asked 2021-03-13 18:28:36 +0100

lsage gravatar image

updated 2021-03-13 18:31:24 +0100

I have the following elliptic curves defined over $\mathbb{C}$ depending on a parameter $a \in \mathbb{C}$:

$$ E_1: y^2 + xy = x^3 + ax + a $$ $$ E_2: y^2 + xy = x^3 + \frac{1}{a}x + \frac{1}{a} $$

That translated to code should look like this:

QQbar_a = QQbar['a']

E1 = EllipticCurve(QQbar_a, [1, 0, 0, a, a])

E2 = EllipticCurve(QQbar_a, [1, 0, 0, 1/a, 1/a])

While defining $E_1$ works fine, defining $E_2$ returns the following error:

TypeError: fraction must have unit denominator

I do not understand if the problem is that I have not specified that $a \neq 0$ (if so, how can I specify it?) or something else. Do you have any guess?

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Closed for the following reason the question is answered, right answer was accepted by lsage
close date 2021-03-13 21:21:02.291740

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answered 2021-03-13 20:36:57 +0100

rburing gravatar image

updated 2021-03-13 20:40:39 +0100

When you write QQbar['a'] you get a polynomial ring over QQbar with variable named a, so an elliptic curve over QQbar['a'] must have coefficients polynomial in a.

To use coefficients like 1/a, you can define instead QQbar_a = QQbar['a'].fraction_field(), or QQbar_a.<a> = FunctionField(QQbar).

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Asked: 2021-03-13 18:28:36 +0100

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Last updated: Mar 13 '21