Timeline for Can elliptic integral singular values generate cubic polynomials with integer coefficients?
Current License: CC BY-SA 3.0
11 events
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| Mar 20, 2015 at 11:26 | history | edited | Wolfgang | CC BY-SA 3.0 |
updated broken links
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| Feb 19, 2012 at 14:51 | vote | accept | Wolfgang | ||
| Feb 19, 2012 at 1:07 | answer | added | Noam D. Elkies | timeline score: 11 | |
| Feb 5, 2012 at 13:16 | comment | added | Wolfgang | Thank you very much. I have edited the lower limit, as it is exactly examples like yours I'm interested in. I suppose the curve is uniquely determined by $d$, up to translation? So again the question: What about $n=-7$? | |
| Feb 5, 2012 at 13:08 | history | edited | Wolfgang | CC BY-SA 3.0 |
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| Feb 5, 2012 at 7:12 | comment | added | Noam D. Elkies | spanferkel's demand that the integral be over $(0,\infty)$ means that the curve must also have a rational 2-torsion point, which cuts the usual list of 13 CM discriminants down to seven: $$-3, -4, -7, -8, -12, -16, -28.$$ For example, the disc.-11 curve yields the complete elliptic integral of $dt / \sqrt{4t^3-4t^2-28t+41}$ which is a rational multiple of $$\pi^{-2} \phantom.\Gamma(1/11)\phantom.\Gamma(3/11)\phantom.\Gamma(4/11) \Gamma(5/11)\phantom.\Gamma(9/11),$$ but the cubic is irreducible so the integral cannot be written as $\int_0^\infty dt/\sqrt P(t)$. | |
| Feb 5, 2012 at 4:27 | comment | added | Will Sawin | If I understand what is going on correctly, any polynomial with the same $j$-invariant should do: en.wikipedia.org/wiki/J-invariant#Algebraic_definition | |
| Feb 5, 2012 at 0:03 | comment | added | Wolfgang | I didn't necessarily think of that restriction. So if we restrict them to that, which polynomial(s) would that be for the $d=-7$ case? And how to find them? Unique? | |
| Feb 4, 2012 at 22:42 | comment | added | Will Sawin | Is this really about elliptic curves with complex multiplication? If so, then the answer is going to be yes if and only if the class number of $\mathbb Q(\sqrt{d})=1$. | |
| Feb 4, 2012 at 22:30 | history | edited | Wolfgang | CC BY-SA 3.0 |
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| Feb 4, 2012 at 22:20 | history | asked | Wolfgang | CC BY-SA 3.0 |