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Finding Cremona label for Tunnell's y^2-x^3-n^2x

asked 2012-07-02 04:20:30 -0500

memory13 gravatar image

When I enter

E = EllipticCurve([-n^2,0]); E;E.label()

Sage gives an error that I have an unsupported operand type. I assume the problem is that I have not defined the variable 'n', which is a square free integer, but how does one do this? Any help would be greatly appreciated!!

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answered 2012-07-02 04:28:33 -0500

kcrisman gravatar image

Correct; n is indeed a builtin thing in Sage, but it's for numerical approximation.

sage: var('n')
n
sage: E = EllipticCurve([-n^2,0]); E
Elliptic Curve defined by y^2 = x^3 + (-n^2)*x over Symbolic Ring

but there is no E.label(), as you'll find if you try it.

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Thank you!

memory13 gravatar imagememory13 ( 2012-07-02 12:15:31 -0500 )edit
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answered 2013-12-09 22:43:44 -0500

John Cremona gravatar image

E.label() only works if E is defined over Q and has conductor within the range of the database (currently up to 300,000, but you have to install an optional spkg to get conductors over 10000).

In your case you wanted to treat n as a variable, so your E is (perhaps) defined over the function field Q(n); such a curve does not have a "Cremona label" -- I should know!

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Haha! Though I bet eventually someone will name something after you that you won't have known of!

kcrisman gravatar imagekcrisman ( 2013-12-10 06:36:01 -0500 )edit

Personally, I am trying to move towards using LMFDB labels instead since there is at least a logical explanation for the ordering used: see http://www.lmfdb.org/knowledge/show/ec.q.lmfdb_label

John Cremona gravatar imageJohn Cremona ( 2013-12-15 03:39:33 -0500 )edit

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Asked: 2012-07-02 04:20:30 -0500

Seen: 177 times

Last updated: Dec 09 '13