# A curiosity for congruent number elliptic curves

I have recently been perfoming a large number of computations on congruent number elliptic curves, using the Birch and Swinnerton-Dyer conjecture to predict heights.

If $$N=8M+7$$ is squarefree, then we expect $$E_N: y^2=x^3-N^2x$$ and the 2-isogenous curve $$F_N:v^2=u^3+4N^2u$$ to have odd rank. For the curves with rank $$1$$, the projected height of the generator of $$F_N$$ is nearly always half that of the generator of $$E_N$$, and so would normally be easier to find.

The exceptions to this all have at least 3 odd prime factors. For those with exactly $$3$$ factors, the differences between the primes are always a multiple of $$4$$.

Is this known? Is it obvious? If not, anybody explain it?

This could not be a comment, so it is an answer.

I'm not sure i got correctly the message from the OP, but here is a statistics of the heights $$h(F)=h(P(F_N))$$, and $$h(E)=h(P(E_N))$$ of generators $$P(F_N)$$ for $$F=F_N$$ and $$P(E_N)$$ for $$E=E_N$$ in the cases where $$h(F). The table was generated using sage, and in the cases where the the rank could not be "easily computed" the case was skipped. So many "bigger heights" do not occur.

The list contains only found cases where the height $$h(E)$$ of the generator of $$E(\Bbb Q)$$ is smaller as the corresponding height $$h(F)$$. The quotient $$h(F)/h(E)$$ is two, shown in the last column.

$$\begin{array}{|c|l||c|c||r|r|c|} \hline N & N & r_F & r_E & h(F) & h(E) & {\displaystyle \frac{h(F)}{h(E)}} \\\hline 1023 & 3 \cdot 11 \cdot 31 & 1 & 1 & 35.8473662698859 & 17.9236831349429 & 2 \\\hline 1239 & 3 \cdot 7 \cdot 59 & 1 & 1 & 26.0698475230467 & 13.0349237615233 & 2 \\\hline 1311 & 3 \cdot 19 \cdot 23 & 1 & 1 & 6.42070898244804 & 3.21035449122402 & 2 \\\hline 1351 & 7 \cdot 193 & 1 & 1 & 18.2620621894512 & 9.13103109472562 & 2 \\\hline 1407 & 3 \cdot 7 \cdot 67 & 1 & 1 & 42.3801890482000 & 21.1900945241000 & 2 \\\hline 1463 & 7 \cdot 11 \cdot 19 & 1 & 1 & 7.91240380926402 & 3.95620190463201 & 2 \\\hline 1551 & 3 \cdot 11 \cdot 47 & 1 & 1 & 20.9996584892411 & 10.4998292446205 & 2 \\\hline 1631 & 7 \cdot 233 & 1 & 1 & 10.2437432584114 & 5.12187162920569 & 2 \\\hline 1679 & 23 \cdot 73 & 1 & 1 & 18.0532459865845 & 9.02662299329226 & 2 \\\hline 1743 & 3 \cdot 7 \cdot 83 & 1 & 1 & 59.8793770087365 & 29.9396885043682 & 2 \\\hline 1751 & 17 \cdot 103 & 1 & 1 & 11.6822329722788 & 5.84111648613939 & 2 \\\hline 1767 & 3 \cdot 19 \cdot 31 & 1 & 1 & 45.8220501212261 & 22.9110250606130 & 2 \\\hline 1967 & 7 \cdot 281 & 1 & 1 & 38.3199662503416 & 19.1599831251708 & 2 \\\hline 2079 & 3^{3} \cdot 7 \cdot 11 & 1 & 1 & 4.33024941358404 & 2.16512470679202 & 2 \\\hline 2159 & 17 \cdot 127 & 1 & 1 & 20.1711247777071 & 10.0855623888536 & 2 \\\hline 2247 & 3 \cdot 7 \cdot 107 & 1 & 1 & 39.9265599852503 & 19.9632799926251 & 2 \\\hline 2343 & 3 \cdot 11 \cdot 71 & 1 & 1 & 30.7338660826787 & 15.3669330413393 & 2 \\\hline 2415 & 3 \cdot 5 \cdot 7 \cdot 23 & 1 & 1 & 11.8334303474177 & 5.91671517370884 & 2 \\\hline 2567 & 17 \cdot 151 & 1 & 1 & 20.7548562007538 & 10.3774281003769 & 2 \\\hline 2607 & 3 \cdot 11 \cdot 79 & 1 & 1 & 29.6073197109170 & 14.8036598554585 & 2 \\\hline 2679 & 3 \cdot 19 \cdot 47 & 1 & 1 & 30.7125711747342 & 15.3562855873671 & 2 \\\hline 2751 & 3 \cdot 7 \cdot 131 & 1 & 1 & 16.7203012594042 & 8.36015062970211 & 2 \\\hline 2807 & 7 \cdot 401 & 1 & 1 & 38.0565222110109 & 19.0282611055054 & 2 \\\hline 2919 & 3 \cdot 7 \cdot 139 & 1 & 1 & 28.5411837845288 & 14.2705918922644 & 2 \\\hline 2967 & 3 \cdot 23 \cdot 43 & 1 & 1 & 50.6070151369618 & 25.3035075684809 & 2 \\\hline 3007 & 31 \cdot 97 & 1 & 1 & 22.5050216483494 & 11.2525108241747 & 2 \\\hline 3135 & 3 \cdot 5 \cdot 11 \cdot 19 & 1 & 1 & 5.63906129855214 & 2.81953064927607 & 2 \\\hline 3143 & 7 \cdot 449 & 1 & 1 & 61.2195290884721 & 30.6097645442361 & 2 \\\hline 3247 & 17 \cdot 191 & 1 & 1 & 30.6211232160319 & 15.3105616080159 & 2 \\\hline 3255 & 3 \cdot 5 \cdot 7 \cdot 31 & 1 & 1 & 6.98070877502530 & 3.49035438751265 & 2 \\\hline 3311 & 7 \cdot 11 \cdot 43 & 1 & 1 & 44.7522224043655 & 22.3761112021827 & 2 \\\hline 3399 & 3 \cdot 11 \cdot 103 & 1 & 1 & 29.4291953315624 & 14.7145976657812 & 2 \\\hline 3423 & 3 \cdot 7 \cdot 163 & 1 & 1 & 52.8327239021380 & 26.4163619510690 & 2 \\\hline 3591 & 3^{3} \cdot 7 \cdot 19 & 1 & 1 & 8.95022228245589 & 4.47511114122794 & 2 \\\hline 3759 & 3 \cdot 7 \cdot 179 & 1 & 1 & 27.1627711409189 & 13.5813855704595 & 2 \\\hline 3791 & 17 \cdot 223 & 1 & 1 & 21.6038597996372 & 10.8019298998186 & 2 \\\hline 3927 & 3 \cdot 7 \cdot 11 \cdot 17 & 1 & 1 & 25.7423670140126 & 12.8711835070063 & 2 \\\hline 3999 & 3 \cdot 31 \cdot 43 & 1 & 1 & 25.1390502760760 & 12.5695251380380 & 2 \\\hline 4047 & 3 \cdot 19 \cdot 71 & 1 & 1 & 59.5381153041493 & 29.7690576520746 & 2 \\\hline 4063 & 17 \cdot 239 & 1 & 1 & 33.6201733123958 & 16.8100866561979 & 2 \\\hline 4071 & 3 \cdot 23 \cdot 59 & 1 & 1 & 32.3583960441806 & 16.1791980220903 & 2 \\\hline 4183 & 47 \cdot 89 & 1 & 1 & 63.1797865950490 & 31.5898932975245 & 2 \\\hline 4191 & 3 \cdot 11 \cdot 127 & 1 & 1 & 44.2977627337341 & 22.1488813668671 & 2 \\\hline 4319 & 7 \cdot 617 & 1 & 1 & 13.3164886659964 & 6.65824433299818 & 2 \\\hline 4431 & 3 \cdot 7 \cdot 211 & 1 & 1 & 13.4795492212960 & 6.73977461064800 & 2 \\\hline 4439 & 23 \cdot 193 & 1 & 1 & 25.2379598322772 & 12.6189799161386 & 2 \\\hline 4471 & 17 \cdot 263 & 1 & 1 & 29.1293039942451 & 14.5646519971226 & 2 \\\hline 4487 & 7 \cdot 641 & 1 & 1 & 39.3379697871404 & 19.6689848935702 & 2 \\\hline 4503 & 3 \cdot 19 \cdot 79 & 1 & 1 & 51.3601232550635 & 25.6800616275318 & 2 \\\hline 4543 & 7 \cdot 11 \cdot 59 & 1 & 1 & 82.5900881429937 & 41.2950440714969 & 2 \\\hline 4559 & 47 \cdot 97 & 1 & 1 & 29.6479192329371 & 14.8239596164685 & 2 \\\hline 4607 & 17 \cdot 271 & 1 & 1 & 17.2922821771960 & 8.64614108859801 & 2 \\\hline 4623 & 3 \cdot 23 \cdot 67 & 1 & 1 & 73.9689543233940 & 36.9844771616970 & 2 \\\hline 4711 & 7 \cdot 673 & 1 & 1 & 25.5153821122245 & 12.7576910561122 & 2 \\\hline 4767 & 3 \cdot 7 \cdot 227 & 1 & 1 & 84.7781303084486 & 42.3890651542243 & 2 \\\hline 4807 & 11 \cdot 19 \cdot 23 & 1 & 1 & 74.4648233907134 & 37.2324116953567 & 2 \\\hline 4935 & 3 \cdot 5 \cdot 7 \cdot 47 & 1 & 1 & 7.13641666234485 & 3.56820833117243 & 2 \\\hline 4983 & 3 \cdot 11 \cdot 151 & 1 & 1 & 78.8704905412520 & 39.4352452706260 & 2 \\\hline 5159 & 7 \cdot 11 \cdot 67 & 1 & 1 & 21.7345051397385 & 10.8672525698693 & 2 \\\hline 5183 & 71 \cdot 73 & 1 & 1 & 83.4770516835033 & 41.7385258417516 & 2 \\\hline 5271 & 3 \cdot 7 \cdot 251 & 1 & 1 & 36.9683989599999 & 18.4841994800000 & 2 \\\hline 5359 & 23 \cdot 233 & 1 & 1 & 25.2609677731502 & 12.6304838865751 & 2 \\\hline 5487 & 3 \cdot 31 \cdot 59 & 1 & 1 & 52.1715444480934 & 26.0857722240467 & 2 \\\hline 5511 & 3 \cdot 11 \cdot 167 & 1 & 1 & 37.1021207472074 & 18.5510603736037 & 2 \\\hline 5719 & 7 \cdot 19 \cdot 43 & 1 & 1 & 10.9849752965575 & 5.49248764827874 & 2 \\\hline 5727 & 3 \cdot 23 \cdot 83 & 1 & 1 & 49.3970928745805 & 24.6985464372902 & 2 \\\hline 5767 & 73 \cdot 79 & 1 & 1 & 51.2908998150279 & 25.6454499075139 & 2 \\\hline 5775 & 3 \cdot 5^{2} \cdot 7 \cdot 11 & 1 & 1 & 4.33024941358404 & 2.16512470679202 & 2 \\\hline 5871 & 3 \cdot 19 \cdot 103 & 1 & 1 & 8.95074847677046 & 4.47537423838523 & 2 \\\hline 5943 & 3 \cdot 7 \cdot 283 & 1 & 1 & 95.4948577152046 & 47.7474288576023 & 2 \\\hline 5983 & 31 \cdot 193 & 1 & 1 & 82.3684742087008 & 41.1842371043504 & 2 \\\hline \end{array}$$ (The search was done for $$N$$ in the range $$1000\le N\le 6000$$.)

There are some cases with $$N$$ having two or four prime divisors. Some other patterns using bigger $$N$$ values:

$$\begin{array}{|c|l||c|c||r|r|c|} \hline N & N & r_F & r_E & h(F) & h(E) & {\displaystyle \frac{h(F)}{h(E)}} \\\hline 1000167 & 3 \cdot 7 \cdot 97 \cdot 491 & 1 & 1 & 35.1072424051541 & 17.5536212025771 & 2 \\\hline 1000239 & 3 \cdot 29 \cdot 11497 & 1 & 1 & 18.0349083276430 & 9.01745416382151 & 2 \\\hline \end{array}$$

In the last case, the situation is as follows. We have $$N=1000239 = 3 \cdot 29 \cdot 11497$$. (The difference $$29-3$$ is not a multiple of four.) The $$2$$-isogenies between $$E_N$$ and $$F_N$$ map the generators as follows:

sage: N = 1000239
sage: EN, FN = EllipticCurve([-N^2, 0]), EllipticCurve([4*N^2, 0])
sage: PEN, PFN = EN.gens()[0], FN.gens()[0]

sage: PEN.xy()
(-5569330752/5929, -152072664682392/456533)
sage: PFN.xy()
(47717344249/379456, -166145928464553715/233744896)

sage: factor(5929), factor(379456)
(7^2 * 11^2, 2^6 * 7^2 * 11^2)

sage: phi_EF(PEN)
(47717344249/379456 : -166145928464553715/233744896 : 1)
sage: phi_EF(PEN) == PFN
True


So the isogeny from $$E_N$$ to $$F_N$$ of degree two maps the generator $$P(E_N)$$ into the other generator, $$P(E_N)\to P(F_N)$$. And if we try to go the other way $$P(F_N)\to 2P(E_N)$$.

sage: FN.isogeny(FN.torsion_points())
Isogeny of degree 2
from Elliptic Curve defined by y^2 = x^3 + 4001912228484*x over Rational Field
to   Elliptic Curve defined by y^2 = x^3 - 16007648913936*x over Rational Field

sage: FN.isogeny(FN.torsion_points())(PFN)
(4376572011592225/136983616 : 287255842000594605842255/1603256241664 : 1)
sage: 2*PEN
(4376572011592225/547934464 : 287255842000594605842255/12826049933312 : 1)


(The codomain of the above isogeny constructed by declaring the kernel $$F_N[2]$$, is not exactly $$E_N$$, but an isomorphich curve, and

sage: (2*PEN).height()
36.0698166552861
sage: FN.isogeny(FN.torsion_points())(PFN).height()
36.0698166552861


the corresponding heights are equal.)