Timeline for Integer solutions of an algebraic equation
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 1, 2023 at 10:06 | vote | accept | Fedor Nilov | ||
| Jan 17, 2023 at 2:11 | comment | added | Joachim König | By the way, I didn't see the reason for this during the computation, but after all is done, LMFDB database says that this elliptic curve is the modular curve $X_1(14)$, which (away from cusps) parameterizes elliptic curves with a $14$-torsion point. Since Mazur's theorem tells us there are no elliptic curves with rational $14$-torsion point, that's one reason why the rank turns out to be $0$. | |
| Jan 17, 2023 at 1:53 | comment | added | Joachim König | @FedorNilov The equation fulfilled by $x$ and $y$ already happens to be quadratic in $y$, namely $y^2+\frac{1}{4}(x-3)(x-1)(x+1)y-\frac{1}{16}(x-1)^4(x+1)^2/x = 0$. From here, it's really an easy exercise to get a Weierstrass model: complete the square and scale $y$ by some suitable expression in $x$ to get (with the thus shifted variable called $y'$): $(y')^2 = x(x+1)(x^2-3x+4)$; if you want to get to $Y^2=f(X)$ with $f$ cubic, replacing $x$ by $1/x$ (to switch $0$ and $\infty$), and doing some further ``cosmetics" gives, e.g., $Y^2=X(X^2-11X+32)$, with all coordinate changes easy to follow. | |
| Jan 16, 2023 at 19:54 | comment | added | Fedor Nilov | Does Magma give an explicit change of variables formula (converting a curve to Weierstrass form)? | |
| Jan 16, 2023 at 14:08 | comment | added | Joachim König | @FedorNilov Magma has functions to compute Weierstrass forms of genus 1 curves (when given one rational point), and also for returning the "Cremona label" of the given elliptic curve; I was lazy and just followed the machine here - although at least a Weierstrass model could have been obtained by hand given enough time. | |
| Jan 16, 2023 at 11:35 | comment | added | Fedor Nilov | At first, you take c=1. Then you consider substitution x=a+b, y=ab and obtain a curve of degree 6. How do you obtain the curve of the weierstrass form here lmfdb.org/EllipticCurve/Q/14/a/5 ? | |
| Jan 16, 2023 at 9:43 | vote | accept | Fedor Nilov | ||
| Jan 16, 2023 at 10:14 | |||||
| Jan 15, 2023 at 13:02 | history | edited | Joachim König | CC BY-SA 4.0 |
edited body
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| Jan 15, 2023 at 12:42 | history | edited | Joachim König | CC BY-SA 4.0 |
added 69 characters in body
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| Jan 15, 2023 at 11:49 | history | answered | Joachim König | CC BY-SA 4.0 |