Ask Your Question
0

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

asked 2021-03-13 18:28:36 +0200

lsage gravatar image

updated 2021-03-13 18:31:24 +0200

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']
QQbar_a.inject_variables()

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

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

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?

edit retag flag offensive reopen merge delete

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

1 Answer

Sort by ยป oldest newest most voted
1

answered 2021-03-13 20:36:57 +0200

rburing gravatar image

updated 2021-03-13 20:40:39 +0200

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).

edit flag offensive delete link more

Question Tools

1 follower

Stats

Asked: 2021-03-13 18:28:36 +0200

Seen: 31 times

Last updated: Mar 13