The elliptic curve $y^2 = 23328x^3-890273x^2+14755570x-7^7$

Question

The elliptic curve,

$$y^2 = 23328x^3-890273x^2+14755570x-7^7 \tag{1}$$

has the small solution $x = 58$. I know how to find other rational points, but the number of digits in the denominator gets large fast.

Question: Does (1) have other rational points of small height, maybe where the numerator or denominator has only 5 digits or less?

P.S. I routinely come across situations like this. Is there an online Alpertron equivalent for elliptic curves, where given $ax^4+bx^3+cx^2+dx+e = y^2$, you just input {$a,b,c,d,e$} into the applet, and it outputs, if any, rational "x" with small height below a bound? (The Alpertron is a very useful site.)

Answer 1

The answer to your question is no, there are no other points with numerator or denominator of $5$ digits or less.

If you have access to a Linux system, you may want to try Michael Stoll's "ratpoints", which can be found here. The documentation for ratpoints, and a description of the algorithm is here. This program tries to find all rational points within a given height bound on a hyperelliptic curve in the most efficient way possible.

Otherwise, ratpoints can be accessed through Sage. In the Sage command line, type

from sage.libs.ratpoints import ratpoints

Then,

ratpoints([$a_0$,$a_1$,$a_2$,...,$a_n$], H, verbose=False, max=0)

finds all the rational points on $y^2=a_0+a_1x+\cdots+a_nx^n$, where H is the bound for the denominator and the absolute value of the numerator of the x-coordinate. When I type,

ratpoints([-7^7,14755570,-890273,23328],10000000)

the answer is

[(1, 0, 0), (58, 49109, 1), (58, -49109, 1), (5170922, 1182208159673289, 344763), (5170922, -1182208159673289, 344763)]

meaning that after $\pm P =(58,\pm 49109)$ the next point with lowest height is

$$\pm Q =\left(\frac{5170922}{344763} , \pm \frac{387482189339}{38958219}\right),$$

and there are no other points (other than $\pm P$ or $\pm Q$) such that the $x$ coordinate has a numerator (in absolute value) or denominator less than $10^7$.

Answer 2

There is John Cremona's mwrank. A more friendly implementation comes with SAGE. Expect a steep learning curve.

Answer 3

Another approach to find all solutions to this elliptic curve would be to determine the full Mordell-Weil group of rational points, using a $2$-descent.

First, one would have to find a minimal model in Weierstrass form for your curve. This is easy, here is one:

$$E : y^2 = x^3 + x^2 + 80022598784x + 49433175273149108$$

But the conductor of this curve is big:

$$N_E =3020151629712 = 2^4\cdot 3 \cdot 7 \cdot 3947 \cdot 2277311,$$

and the $2$-descent is a very lengthy process. I had my computer running for a while, trying to calculate the rank (or just some bounds for the rank), but it didn't get anywhere. Maybe more patient users can let their computer run and see if they can find generators for the group of rational points of this elliptic curve.