Help to find cubics with square discriminant

Question

If the discriminant $b^2-4c$ of the quadratic $x^2 + bx + c$ is a square then it factors. For every discriminant $d^2$ we have can parametrize them all $(b,c) = (d + 2 h,h(d+h))$. edit I realized now that the quadratic case is trivial because it the discriminant is a square iff it factors, so the two variable parametrization is $(x-a)(x-b)$ so it might not represent what is happening with the cubic.

I was hoping to do a similar parametrization for the cubics $x^3 - ax + b$ with square discriminant $4a^3 - 27b^2 = d^2$ but factoring in the Eisenstein integers does not seem to make the problem any easier.

Are there any other promising approaches I could try?

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I noticed the problem is simple when $d=0$. In that case we have $a = 3 m^2$, $b = 2 m^3$. Also when $d = b$ we also get a simple parametrization, $a = 7m^2$, $b = 7m^3$ but I don't think these will help to get the general case.

Answer 1

An equation like $4a^3−27b^2=d^2$ for fixed $d$ defines a curve in the plane with coordinates $a,b$. It can be parametrised by rational functions if and only if it is singular, which is the case in your examples $d = 0$ and $d = b$. If not, it can be completed to an elliptic curve, which can not be parametrised by rational functions.

Answer 2

If you factor in the Eisenstein integers, you can get a parametrization though it's a bit messy.

Write $b + 3\sqrt{-3}d = b + 3d(2j+1) = (b+3d)+(3d)j = 2u$, so you can write $a^3 = u \bar{u}$. Since $u \bar{u}$ must be a cube, $u$ has to factor into bunches of product of primes of the form $p^3$ or $p^2 \bar{p}$, so you can find Eisenstein integers $v$ and $w$ such that $u = v^3w^2\bar{w}$ (though this decomposition is generally not unique)

This parametrization gives you $a = vw\bar{v}\bar{w}$ and $(b+3d)+(3d)j = 2v^3w^2\bar{w}$. If you want to get integer parameters, then you will have 4 parameters, and in order to get the equations for $b$ and $d$ you have to expand the product $v^3w^2\bar{w}$ to get its integer components, and then solve the system for $b$ and $d$.

Answer 3

I would not focus on parametrization as such. Instead, note that $4 a^3 = d^2 + 27 b^2.$ This gives an easily stated restriction on the prime factorization of $a.$ It is necessary and sufficient that $a \geq 0$ and, whenever any prime $p | a$ and $p \equiv 2 \pmod 3,$ then the exponent of $p$ must be even. So $a = 2$ or $a = 5$ or $a=10$ are impossible. Without the cube, there would be a restriction on the prime 3 as well, but it turns out not to matter because you have $a^3.$ Without the $4$ in $4a^3,$ there would be a competition between $d^2 + 27 b^2$ and the other forms in the genus, $4 u^2 \pm 2 u v + 7 v^2.$ But $$ \left( \begin{array}{cc} 4 & 0 \\ 1 & 1 \end{array} \right) \; \cdot \; \left( \begin{array}{cc} 1 & 0 \\ 0 & 27 \end{array} \right) \; \cdot \; \left( \begin{array}{cc} 4 & 1 \\ 0 & 1 \end{array} \right) = \; \;\; \; \left( \begin{array}{cc} 16 & 4 \\ 4 & 28 \end{array} \right) $$

Now, try $a = 4 = 2^2,$ so the exponent on 2 is even, you get $4a^3 = 256,$ and you get $d = 16, b = 0,$ which seems too easy. i will put some more below, maybe skip squares... But, given a legal $a$ as described, the number of pairs $d,b$ is finite.

$$a = 1,4a^3 = 4, ( d = \pm 2, b = 0), $$ $$a = 3, 4a^3 = 108, ( d = \pm 9, b = \pm 1), ( d = 0, b = \pm 2), $$ $$a = 7, 4a^3 = 1372, ( d = \pm 20, b = \pm 6), ( d = \pm 7, b = \pm 7), $$ $$a = 12, 4a^3 = 6912, ( d = \pm 72, b = \pm 8), ( d = 0, b = \pm 16), $$ $$a = 13, 4a^3 = 8788, ( d = \pm 70, b = \pm 12), ( d = \pm 65, b = \pm 13). $$

Well, given all possible representations $a = s^2 + s t + t^2,$ one may construct all $d,b,$ an annoying task unless $a$ is prime with $a \equiv 1 \pmod 3.$ This is, essentially, what factoring in the Eisenstein integers will give you.

Actually, that last bit was needlessly pessimistic. With a little special treatment of the primes 2,3, I can see how to create all possible representations of $a = j^2 + 3 k^2,$ create all possible representations of $4a^3 = m^2 + 3 n^2,$ then just keep the ones when $3 | n.$ Rather involved but easy enough to program.

Answer 4

The discriminant of an irreducible cubic is a square if and only if it is a cyclic cubic, that is, if and only if its Galois group is the cyclic group of order $3$. There are tabulations in the literature of all cyclic cubics with discriminant less than $n$, for various values of $n$. These can be found by web-searching the phrase "cyclic cubic" (or "abelian cubic", which amounts to the same thing, since if the Galois group isn't the cyclic group of order $3$, it must be the symmetric group on three letters, which group is not abelian).

Answer 5

The discriminant of the cubic $x^3 - a x + b$ is $4 a^3 - 27 b^2$ and we are looking for triplets $(a,b,c)$ satisfying

$$4 a^3 = 27 b^2 + c^2= F(b,c)$$ or $$a = 27 \left(\frac{b}{2a}\right)^2 + \left(\frac{c}{2 a}\right)^2$$

so we can parametrize rational solutions $(a,b,c)$ by

\begin{eqnarray} a &=& F(u,v) \\ b &=& 2 u F(u,v) \\ c &=& 2 v F(u,v) \end{eqnarray}

where $F(u,v) = 27 u^2 + v^2$. Therefore, the reduced cubics with square discriminant are of the form

$$P(x,u,v)=x^3 - (27 u^2 + v^2) x + 2 u(27 u^2 + v^2)$$

and the general cubics with square discriminant( $\Delta = (2 v(27 u^2 + v^2))^2$ ) are of the form

$$P(x+w, u,v) = x^3 + 3 w x^2 + (3 w^2 -(27 u^2 + v^2))x + ( w^3 - (27 u^2 + v^2) w + 2u ( 27 u^2 + v^2))$$

$\bf{Added:}$ Reduced cubics with square determinant can also be obtained as

$$x^3 - 3(u^2 - u v + v^2) x + (u^3 + v^3)$$ with discriminant $\Delta= 81(u-v)^2(u^2 - u v + v^2)^2$

It is not hard to check that we get all reduced cubics with rational coefficients and square discriminant in this way. This implies no such cubics have constant term $\pm 1$.