Finding the maximum area of isosceles triangle

A curiosity for congruent quantity elliptic curves Answer

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A curiosity for congruent quantity elliptic curves

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I’m not positive i bought accurately the message from the OP, however here’s a statistics of the heights $h(F)=h(P(F_N))$, and $h(E)=h(P(E_N))$ of turbines $P(F_N)$ for $F=F_N$ and $P(E_N)$ for $E=E_N$ within the circumstances the place $h(F)<h(E)$. The desk was generated utilizing anecdote, and within the circumstances the place the the rank couldn’t breathe “easily computed” the illustration was skipped. So many “bigger heights” don’t happen.

The listing incorporates solely create circumstances the place the peak $h(E)$ of the generator of $E(Bbb Q)$ is smaller because the corresponding peak $h(F)$. The quotient $h(F)/h(E)$ is 2, proven within the final column.

$$
commence{array}r
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
aim{array}
$$

(The search was achieved for $N$ within the meander $1000le Nle 6000$.)

There are some circumstances with $N$ having two or 4 prime divisors.
Some different patterns utilizing larger $N$ values:

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

In the final illustration, the status is as follows.
We have $N=1000239 = 3 cdot 29 cdot 11497$. (The dissimilarity $29-3$ shouldn’t be a a number of of 4.) The $2$-isogenies between $E_N$ and $F_N$ map the turbines as follows:

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

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

anecdote: issue(5929), issue(379456)
(7^2 * 11^2, 2^6 * 7^2 * 11^2)

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

So the isogeny from $E_N$ to $F_N$ of diploma two maps the generator $P(E_N)$ into the opposite generator, $P(E_N)to P(F_N)$. And if we attempt to refer the opposite route $P(F_N)to 2P(E_N)$.

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

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

(The codomain of the above isogeny constructed by declaring the kernel $F_N[2]$, shouldn’t be precisely $E_N$, however an isomorphich round, and

anecdote: (2*PEN).peak()
36.0698166552861
anecdote: FN.isogeny(FN.torsion_points())(PFN).peak()
36.0698166552861

the corresponding heights are equal.)

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