transformation of singular quartics into Weierstrass form

Question

I need a birational transformation of singular quartics into weierstrass form. the quartics are of genus one and have the following form: $$(x^2 + c) (y^2 + i) + k = 0$$ where $c,i,k$ are given rational numbers, and $k$ is of big height. The only known rational solutions: the singular points in homogenous coordinates: $(x,y,z)=(0,1,0),(1,0,0)$.

Methods described in the literature (e.g. Lawrence Washington, Elliptic curves, p.37) require a non-singular rational solution. but after this text a singular quartic can be transformed birationally into a nonsingular curve, where the singular solutions are mapped into two nonsingular solutions. Unfortunately, I have not found in this work and in google, how to do this transformation. I ask for information, how to transform singular quartics as above into Weierstrass form or how to transform into a nonsingular curve.

Answer 1

This is just an extened comment. I would conjecture that it is unlikely that there is a nice closed from for a Weierstrass equation for the curve $$C: (X^2 + rZ^2)(Y^2 + sZ^2) + tZ^4 = 0$$

over the function field $\mathbb{Q}(r,s,t)$ - maybe some expert on elliptic surfaces will come along and say that this is codswhallop.

Why do I say this? Let's ask Magma about the case where $r = -u^2$ (we will see why this is helpful). Take the affine piece where $y = 1$ and compute the blowup (as in e.g., Hartshorne I Example 4.9.1) see that the curve $$C' : sx^2z^2 + x^2 + (rs + t)z^2 + r = 0$$ is birational to $C$. Notice that $C'$ has the nonsingular point $(u, 0)$.

Now let's ask Magma what the Weierstrass equation is, shall we?

K<r,s,t> := FunctionField(Rationals(), 3);

A2<x,z> := AffineSpace(K, 2);

f := (x^2 -r^2*z^2)*(1 + s*z^2) + t*z^4; 
C:= Curve(A2, f);

g := ExactQuotient(Evaluate(f, [x*z, z]), z^2);

C1 := Curve(A2, g);
C1_proj := ProjectiveClosure(C1);
P := C1_proj![r, 0, 1];
IsSingular(P);                            //Should be false for the next bit to work

E := EllipticCurve(C1_proj, P);
WeierstrassModel(E);

Which outputs the lovely equation

Elliptic Curve defined by y^2 = x^3 + (-6912*r^8*s^6 + 6912*r^6*s^5*t -
    432*r^4*s^4*t^2)/(r^16*s^8 - 6*r^14*s^7*t + 31/2*r^12*s^6*t^2 -
    45/2*r^10*s^5*t^3 + 321/16*r^8*s^4*t^4 - 45/4*r^6*s^3*t^5 + 31/8*r^4*s^2*t^6
    - 3/4*r^2*s*t^7 + 1/16*t^8)*x + (-221184*r^10*s^8 + 221184*r^8*s^7*t +
    6912*r^6*s^6*t^2)/(r^22*s^11 - 17/2*r^20*s^10*t + 65/2*r^18*s^9*t^2 -
    295/4*r^16*s^8*t^3 + 1765/16*r^14*s^7*t^4 - 3653/32*r^12*s^6*t^5 +
    667/8*r^10*s^5*t^6 - 1375/32*r^8*s^4*t^7 + 245/16*r^6*s^3*t^8 -
    115/32*r^4*s^2*t^9 + 1/2*r^2*s*t^10 - 1/32*t^11) over Multivariate rational
function field of rank 3 over Rational Field