Why are squares sometimes included in quadratic twists

[Maarten Derickx]

On https://www.lmfdb.org/EllipticCurve/Q/64/a/1 there is the text

Twists

The minimal quadratic twist of this elliptic curve is 32.a1, its twist by 8.

Is there a good reason why to write the twist by 8 and not the twist by 2? I think it would be more elegant if the number mentioned there would always be squarefree.

Thanks,

Maarten

[Maarten Derickx]

I first thought I found the answer, because 8 is the discriminant of the quadratic field Q(sqrt(2)) and the normalization is to use fundamental discriminants. But this is not the case, since at https://www.lmfdb.org/EllipticCurve/Q/32/a/2 there is the text

Twists

The minimal quadratic twist of this elliptic curve is 32.a1, its twist by −1.

And the discriminant of Q(sqrt(-1)) is -4. 

[Maarten Derickx]

p.s. I realized there were some incorrect links on the relevant knowl https://beta.lmfdb.org/knowledge/show/ec.q.minimal_twist which I fixed in beta.

[Andrew Sutherland]

Thanks for the bug report, the database has been updated so that "minimal twist by" always lists the discriminant of the quadratic field over which the curves are isomorphic (which was always the intention, the inconsistency was inadvertent).  Thank you for letting us know, and thank you for fixing the broken links in the knowl.

Best,

Drew

[John Cremona]

I looked to see how this happened, to make sure that for any new data we are consistent.  I found that the code I used in May to make the min quad twists consistent with our new definition (1) made no changes when the existing min quad twist curve was correct, and (2) when the curve changed, put in the column 'min_quad_twist_disc' the scond thing output by Sage's method E.is_quadratic_twist(), which is (as documented) a square-free integer.  So the non-square-free ones Maarten saw must have been there from before, for curves where the new work did not change the curve.

Since our new policy is to show the discriminant rather than its squarefree part, I will change my script for future use.

John

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[cafe wilson]

Maarten Derickx <m.derick...@gmail.com> 於 2023年8月27日週日 上午5:10寫道：

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