rank between two elliptic curves

Question

Given an elliptic curve $E:\ y^2 =x^3 + ax + b$, and another curve $F:\ y^2 = x^3 + ax - b$. What is the relation between rank $E(\Bbb Q)$ and rank $F(\Bbb Q)$? are there any specific transformation between $E(\Bbb Q)$ and $F(\Bbb Q)$?

Answer 1

Over $\Bbb Q$ there is no relation between the two ranks. Here is a small piece of sage code that checks all cases of differences for values of $a$ between $-4$ and $4$, and of $b$ between $1$ and $10$. (Of course, for $b=0$ we have an empty problem.)

for a in [-4..4]:
    for b in [1..8]:
        rE, rF = -1, -1
        try:
            E = EllipticCurve(QQ, [a, +b])
            F = EllipticCurve(QQ, [a, -b])
        except:
            continue
        rE = E.rank()
        rF = F.rank()
        if rE == rF:    # alternatively if abs(rE - rF) < 3:
            continue
        print(f'{a} & {b} & {rE} & {rF} \\\\\\hline')

The result is printed above so that it can be included in the following latex / mathjax array block: $$ \begin{aligned} E&: &y^2 &=x^3 + ax + b\ ,\\ F&: &y^2 &=x^3 + ax - b\ ,\\ \end{aligned} $$ $$ \begin{array}{|r|r||r|r|} \hline a & b & \operatorname{rank} E(\Bbb Q) & \operatorname{rank} F(\Bbb Q) \\\hline -4 & 1 & 2 & 0 \\\hline -4 & 2 & 0 & 1 \\\hline -4 & 4 & 1 & 0 \\\hline -4 & 5 & 0 & 1 \\\hline -4 & 7 & 1 & 0 \\\hline -3 & 6 & 1 & 0 \\\hline -3 & 7 & 1 & 0 \\\hline -2 & 2 & 1 & 0 \\\hline -2 & 4 & 1 & 0 \\\hline -2 & 6 & 1 & 0 \\\hline -2 & 7 & 0 & 1 \\\hline -2 & 8 & 2 & 1 \\\hline -1 & 1 & 1 & 0 \\\hline -1 & 2 & 0 & 1 \\\hline -1 & 3 & 1 & 0 \\\hline -1 & 4 & 2 & 0 \\\hline -1 & 5 & 0 & 1 \\\hline -1 & 7 & 1 & 0 \\\hline -1 & 8 & 0 & 1 \\\hline 0 & 3 & 1 & 0 \\\hline 0 & 4 & 0 & 1 \\\hline 0 & 5 & 1 & 0 \\\hline 0 & 7 & 0 & 1 \\\hline 0 & 8 & 1 & 0 \\\hline 1 & 3 & 1 & 0 \\\hline 1 & 5 & 0 & 1 \\\hline 2 & 1 & 1 & 0 \\\hline 2 & 2 & 0 & 1 \\\hline 2 & 4 & 2 & 0 \\\hline 2 & 6 & 1 & 0 \\\hline 2 & 7 & 1 & 0 \\\hline 2 & 8 & 0 & 2 \\\hline 3 & 1 & 1 & 0 \\\hline 3 & 2 & 1 & 0 \\\hline 3 & 3 & 0 & 1 \\\hline 3 & 4 & 1 & 0 \\\hline 3 & 6 & 0 & 1 \\\hline 3 & 7 & 0 & 1 \\\hline 3 & 8 & 1 & 0 \\\hline 4 & 1 & 1 & 2 \\\hline 4 & 3 & 0 & 1 \\\hline 4 & 7 & 0 & 1 \\\hline \end{array} $$ The cases above already show too many discrepancies. Very often one rank is zero and the other is not. (So how to use the information from the rank zero curve to exhibit a generator for the other curve?!)

---

Here are some further examples (with bigger rank differences), obtained in a search while letting $b$ run till thousand instead, and printing only those cases where the rank difference is at least $3$. (Change the code so that abs(rE - rF) < 3 instead of rE == rF for the case when we continue, and thus do not print the pair $(a,b)$ and the corresponding ranks).

(Some rank computations may have been abandoned when the first try to get the rank / to get easy generators did not work.) $$ \begin{array}{|r|r||r|r|} \hline a & b & \operatorname{rank} E(\Bbb Q) & \operatorname{rank} F(\Bbb Q) \\\hline -4 & 49 & 3 & 0 \\\hline -4 & 64 & 3 & 0 \\\hline -4 & 196 & 3 & 0 \\\hline -4 & 225 & 3 & 0 \\\hline -4 & 256 & 3 & 0 \\\hline -4 & 290 & 0 & 3 \\\hline -4 & 331 & 3 & 0 \\\hline -4 & 361 & 3 & 0 \\\hline -4 & 431 & 0 & 3 \\\hline -4 & 481 & 3 & 0 \\\hline -4 & 484 & 3 & 0 \\\hline -4 & 529 & 3 & 0 \\\hline -4 & 561 & 3 & 0 \\\hline -4 & 571 & 3 & 0 \\\hline -4 & 676 & 4 & 1 \\\hline -4 & 839 & 0 & 3 \\\hline -4 & 841 & 3 & 0 \\\hline -4 & 889 & 3 & 0 \\\hline -4 & 896 & 0 & 3 \\\hline -4 & 921 & 3 & 0 \\\hline -4 & 964 & 3 & 0 \\\hline -3 & 399 & 3 & 0 \\\hline -2 & 81 & 3 & 0 \\\hline -2 & 246 & 3 & 0 \\\hline -2 & 288 & 3 & 0 \\\hline -2 & 497 & 3 & 0 \\\hline -2 & 505 & 3 & 0 \\\hline -2 & 785 & 4 & 1 \\\hline -2 & 844 & 3 & 0 \\\hline -2 & 937 & 3 & 0 \\\hline -2 & 956 & 3 & 0 \\\hline -2 & 998 & 3 & 0 \\\hline -1 & 25 & 3 & 0 \\\hline -1 & 49 & 3 & 0 \\\hline -1 & 64 & 3 & 0 \\\hline -1 & 121 & 3 & 0 \\\hline -1 & 169 & 3 & 0 \\\hline -1 & 190 & 3 & 0 \\\hline -1 & 206 & 0 & 3 \\\hline -1 & 225 & 3 & 0 \\\hline -1 & 236 & 0 & 3 \\\hline -1 & 256 & 3 & 0 \\\hline -1 & 280 & 3 & 0 \\\hline -1 & 335 & 0 & 3 \\\hline -1 & 338 & 0 & 3 \\\hline -1 & 406 & 4 & 1 \\\hline -1 & 417 & 3 & 0 \\\hline -1 & 466 & 3 & 0 \\\hline -1 & 505 & 3 & 0 \\\hline -1 & 506 & 0 & 3 \\\hline -1 & 520 & 3 & 0 \\\hline -1 & 610 & 3 & 0 \\\hline -1 & 619 & 3 & 0 \\\hline -1 & 625 & 4 & 1 \\\hline -1 & 669 & 3 & 0 \\\hline -1 & 695 & 0 & 3 \\\hline -1 & 729 & 4 & 1 \\\hline -1 & 736 & 3 & 0 \\\hline -1 & 753 & 3 & 0 \\\hline -1 & 784 & 3 & 0 \\\hline -1 & 794 & 0 & 3 \\\hline -1 & 865 & 3 & 0 \\\hline -1 & 888 & 0 & 3 \\\hline -1 & 904 & 3 & 0 \\\hline -1 & 941 & 0 & 3 \\\hline -1 & 946 & 3 & 0 \\\hline -1 & 959 & 1 & 4 \\\hline -1 & 961 & 4 & 1 \\\hline 0 & 113 & 3 & 0 \\\hline 0 & 141 & 3 & 0 \\\hline 0 & 568 & 3 & 0 \\\hline 0 & 681 & 3 & 0 \\\hline 0 & 741 & 0 & 3 \\\hline 0 & 745 & 3 & 0 \\\hline 0 & 873 & 3 & 0 \\\hline 0 & 997 & 3 & 0 \\\hline 1 & 293 & 3 & 0 \\\hline 1 & 867 & 3 & 0 \\\hline 2 & 71 & 0 & 3 \\\hline 2 & 97 & 3 & 0 \\\hline 2 & 199 & 3 & 0 \\\hline 2 & 256 & 3 & 0 \\\hline 2 & 361 & 3 & 0 \\\hline 2 & 438 & 4 & 1 \\\hline 2 & 441 & 3 & 0 \\\hline 2 & 496 & 4 & 0 \\\hline 2 & 526 & 3 & 0 \\\hline 2 & 594 & 3 & 0 \\\hline 2 & 609 & 3 & 0 \\\hline 2 & 722 & 0 & 3 \\\hline 2 & 796 & 3 & 0 \\\hline 2 & 998 & 0 & 3 \\\hline 3 & 533 & 3 & 0 \\\hline 3 & 540 & 3 & 0 \\\hline 3 & 838 & 3 & 0 \\\hline 4 & 251 & 3 & 0 \\\hline 4 & 724 & 3 & 0 \\\hline 4 & 769 & 3 & 0 \\\hline 4 & 825 & 3 & 0 \\\hline 4 & 863 & 0 & 3 \\\hline 4 & 977 & 3 & 0 \\\hline \end{array} $$

---

To have a further more striking counterexample, we look inside a precomputed list (sage) of curves of higher rank, say rank five.

sage: E5 = elliptic_curves.rank(5)[0]
sage: E5
Elliptic Curve defined by y^2 + y = x^3 - 79*x + 342 over Rational Field
sage: E = E5.short_weierstrass_model()
sage: E
Elliptic Curve defined by y^2 = x^3 - 1264*x + 21904 over Rational Field
sage: F = EllipticCurve(QQ, (E.a4(), -E.a6()))
sage: F
Elliptic Curve defined by y^2 = x^3 - 1264*x - 21904 over Rational Field
sage: E.is_isomorphic(E5)
True
sage: E5.rank()
5

sage: E.analytic_rank()
5
sage: F.analytic_rank()
1

So we expect ranks $5$ for $E$ (known to be so), and rank $1$ for $F$ with rational point provided by

sage: F.simon_two_descent()
(1, 5, [(985/9 : 28963/27 : 1)])

(but for a proof some further hours of computation may be needed).