This is a partial answer.
Using the group law of elliptic curves, we have
$b_{10}=\frac{\left(n^{102}+133370 n^{100}+235431945 n^{98}-53960558412 n^{96}+37353015835420 n^{94}+1715285459850920 n^{92}+154018873425409932 n^{90}+4528460790052228992 n^{88}-10283455511101671494 n^{86}+424389497721502142548 n^{84}+21198683938074571284634 n^{82}+261033154046082383403576 n^{80}+2270064982003402818480028 n^{78}+15317757618354379225002184 n^{76}+84218098664615395090065004 n^{74}+345159573021345914583822624 n^{72}+1129951624304798874757797847 n^{70}+3131305476125014022272232326 n^{68}+7635830330877839055871498111 n^{66}+16446578072159313295259200380 n^{64}+31726949494863946397847804664 n^{62}+54769575000613973884608095824 n^{60}+84669779180983943334398860888 n^{58}+117776609406232459328970393984 n^{56}+146631440164415842271230869964 n^{54}+164805637943248865339648114968 n^{52}+165741869909257891183520432332 n^{50}+150306415334481456543373768912 n^{48}+121703129810992794049207385688 n^{46}+88124674373459097943143748048 n^{44}+56481814917222827023209545464 n^{42}+31767118902941960531617938880 n^{40}+15407922021660173371718373183 n^{38}+6306682290739982829447936934 n^{36}+2063724483753154237664688919 n^{34}+509509860910406920886925292 n^{32}+72423661674195782496620652 n^{30}-2085965471015373600109304 n^{28}-3102450405774431050519652 n^{26}-511897235486864832178688 n^{24}+72673456634449307534874 n^{22}+33312001105932688908244 n^{20}+3846262561102571988538 n^{18}+294258428803841831032 n^{16}+9380206793092459404 n^{14}+156074955019903848 n^{12}-7836007732260580 n^{10}-12400565501408 n^{8}+607383986505 n^{6}+2125016730 n^{4}-948799 n^{2}+100\right) n}{100 n^{102}-948799 n^{100}+2125016730 n^{98}+607383986505 n^{96}-12400565501408 n^{94}-7836007732260580 n^{92}+156074955019903848 n^{90}+9380206793092459404 n^{88}+294258428803841831032 n^{86}+3846262561102571988538 n^{84}+33312001105932688908244 n^{82}+72673456634449307534874 n^{80}-511897235486864832178688 n^{78}-3102450405774431050519652 n^{76}-2085965471015373600109304 n^{74}+72423661674195782496620652 n^{72}+509509860910406920886925292 n^{70}+2063724483753154237664688919 n^{68}+6306682290739982829447936934 n^{66}+15407922021660173371718373183 n^{64}+31767118902941960531617938880 n^{62}+56481814917222827023209545464 n^{60}+88124674373459097943143748048 n^{58}+121703129810992794049207385688 n^{56}+150306415334481456543373768912 n^{54}+165741869909257891183520432332 n^{52}+164805637943248865339648114968 n^{50}+146631440164415842271230869964 n^{48}+117776609406232459328970393984 n^{46}+84669779180983943334398860888 n^{44}+54769575000613973884608095824 n^{42}+31726949494863946397847804664 n^{40}+16446578072159313295259200380 n^{38}+7635830330877839055871498111 n^{36}+3131305476125014022272232326 n^{34}+1129951624304798874757797847 n^{32}+345159573021345914583822624 n^{30}+84218098664615395090065004 n^{28}+15317757618354379225002184 n^{26}+2270064982003402818480028 n^{24}+261033154046082383403576 n^{22}+21198683938074571284634 n^{20}+424389497721502142548 n^{18}-10283455511101671494 n^{16}+4528460790052228992 n^{14}+154018873425409932 n^{12}+1715285459850920 n^{10}+37353015835420 n^{8}-53960558412 n^{6}+235431945 n^{4}+133370 n^{2}+1}$
All other $b_{m}$ can also be generated. Given,
$$(a^3-b)(b^3-a) = y^2$$
First family ($a=n$)
Denote $$E_1=\{(U,V): V^2 = -U^4+n^3U^3+nU-n^4\}\cup O.$$ It is birationally equivalent to Weierstrass form $$E_2=\{(X,Y): Y^{2}+\left(3 n^{2}-1\right)XY + \left(2 n^{6}-10 n^{4}+8 n^{2}\right)Y = X^{3}+\left(\frac{3}{4} n^{4}-\frac{9}{2} n^{2}-\frac{1}{4}\right) X^{2}+4 n^{2} \left(n -1\right)^{2} \left(n +1\right)^{2} X +3 n^{10}-24 n^{8}+38 n^{6}-16 n^{4}-n^{2} \}\cup O$$ by $$\left[U = \frac{-3 n^{7}+21 n^{5}+\left(-4 X -17\right) n^{3}+\left(4 X +2 Y -1\right) n}{2 Y} , V = -\frac{\left(n +1\right) n \left(-3 n^{10}+\left(\frac{9 X}{8}+24\right) n^{8}+\left(-\frac{35 X}{2}+\frac{Y}{4}-38\right) n^{6}+\left(\frac{9}{4} X^{2}+\frac{191}{4} X -\frac{17}{4} Y +16\right) n^{4}+\left(-\frac{27}{2} X^{2}+\frac{1}{2} X -\frac{17}{4} Y +1\right) n^{2}+X^{3}-\frac{3 X^{2}}{4}+\frac{X}{8}+\frac{Y}{4}\right) \left(n -1\right)}{Y^{2}} , X = \frac{n \left(3 U \,n^{4}-n^{5}-4 U \,n^{2}-2 V \,n^{2}+U +2 V +n \right)}{\left(U -n \right)^{2}} , Y = -\frac{3 \left(n +1\right) n \left(n -1\right) \left(-\frac{n^{6}}{3}+2 U \,n^{5}+\left(U^{2}-6\right) n^{4}+\frac{4 \left(5 U -2 V \right) n^{3}}{3}+\left(-6 U^{2}+1\right) n^{2}+2 \left(U +\frac{4 V}{3}\right) n -\frac{U^{2}}{3}\right)}{2 \left(U -n \right)^{3}} \right] .$$ Let $P=\left(3 n^{2}+1, -2 n^{6}-\frac{1}{2} n^{4}-5 n^{2}-\frac{1}{2}\right)\in E_2,$ then the $U$ corresponding to $[m-1]P$ is exactly $b_m$ where $m\geq 1$.
Second family ($a=n^3$)
Denote $$E_3=\{(U,V):V^{2} = n^{9} U^{3}-n^{12}-U^{4}+n^{3} U\}\cup O.$$ It is birationally equivalent to $$E_4=\{(X,Y):Y^{2} = X^{3}+\left(3 n^{10}-6 n^{2}\right) X^{2}+\left(3 n^{20}-15 n^{12}+12 n^{4}\right) X -9 n^{22}+18 n^{14}-9 n^{6} \}\cup O$$ by $$\left[U = \frac{n \left(3 n^{10}-3 n^{2}+X \right)}{X}, V = \frac{3 Y \,n^{11}-3 Y \,n^{3}}{X^{2}}, X = \frac{3 n^{11}-3 n^{3}}{U -n}, Y = \frac{3 V \,n^{11}-3 V \,n^{3}}{\left(U -n \right)^{2}}\right]. $$ Let $Q=(n^{12}-n^{10}+n^{6}+2 n^{2}+1, n^{18}+n^{12}+n^{6}+1) \in E_4,$ then the $U$ corresponding to $[m]Q$ is exactly $b_m$ where $m\geq 1$.
For example $b_1=\frac{\left(n^{6}+n^{4}-2 n^{2}+1\right) n}{n^{6}-2 n^{4}+n^{2}+1}$
$b_2=\frac{n \left(n^{12}+8 n^{10}+10 n^{6}-4 n^{2}+1\right)}{n^{12}-4 n^{10}+10 n^{6}+8 n^{2}+1}$
$b_3=\frac{\left(n^{30}+17 n^{28}-18 n^{26}+101 n^{24}-172 n^{22}+80 n^{20}+282 n^{18}-82 n^{16}-244 n^{14}+282 n^{12}-28 n^{10}-64 n^{8}+101 n^{6}+9 n^{4}-10 n^{2}+1\right) n}{n^{30}-10 n^{28}+9 n^{26}+101 n^{24}-64 n^{22}-28 n^{20}+282 n^{18}-244 n^{16}-82 n^{14}+282 n^{12}+80 n^{10}-172 n^{8}+101 n^{6}-18 n^{4}+17 n^{2}+1} $
$b_4=\frac{n \left(n^{48}+32 n^{46}+552 n^{42}-1088 n^{40}-16 n^{38}+5820 n^{36}+8160 n^{34}+544 n^{32}+6552 n^{30}+18560 n^{28}-4080 n^{26}+23302 n^{24}+8160 n^{22}-9280 n^{20}+6552 n^{18}-1088 n^{16}-4080 n^{14}+5820 n^{12}+32 n^{10}+544 n^{8}+552 n^{6}-16 n^{2}+1\right)}{n^{48}-16 n^{46}+552 n^{42}+544 n^{40}+32 n^{38}+5820 n^{36}-4080 n^{34}-1088 n^{32}+6552 n^{30}-9280 n^{28}+8160 n^{26}+23302 n^{24}-4080 n^{22}+18560 n^{20}+6552 n^{18}+544 n^{16}+8160 n^{14}+5820 n^{12}-16 n^{10}-1088 n^{8}+552 n^{6}+32 n^{2}+1} $
$b_5=\frac{n \left(n^{78}+49 n^{76}-50 n^{74}+2093 n^{72}-9092 n^{70}+7024 n^{68}+71486 n^{66}+146362 n^{64}-221372 n^{62}+120846 n^{60}+1989884 n^{58}-1997248 n^{56}+2411691 n^{54}+2513095 n^{52}-3945398 n^{50}+4925063 n^{48}+1374136 n^{46}-4812512 n^{44}+9246036 n^{42}+3689036 n^{40}-10556008 n^{38}+9246036 n^{36}+2487832 n^{34}-5926208 n^{32}+4925063 n^{30}+3430735 n^{28}-4863038 n^{26}+2411691 n^{24}+1056332 n^{22}-1063696 n^{20}+120846 n^{18}+102298 n^{16}-177308 n^{14}+71486 n^{12}-3476 n^{10}+1408 n^{8}+2093 n^{6}+25 n^{4}-26 n^{2}+1\right)}{n^{78}-26 n^{76}+25 n^{74}+2093 n^{72}+1408 n^{70}-3476 n^{68}+71486 n^{66}-177308 n^{64}+102298 n^{62}+120846 n^{60}-1063696 n^{58}+1056332 n^{56}+2411691 n^{54}-4863038 n^{52}+3430735 n^{50}+4925063 n^{48}-5926208 n^{46}+2487832 n^{44}+9246036 n^{42}-10556008 n^{40}+3689036 n^{38}+9246036 n^{36}-4812512 n^{34}+1374136 n^{32}+4925063 n^{30}-3945398 n^{28}+2513095 n^{26}+2411691 n^{24}-1997248 n^{22}+1989884 n^{20}+120846 n^{18}-221372 n^{16}+146362 n^{14}+71486 n^{12}+7024 n^{10}-9092 n^{8}+2093 n^{6}-50 n^{4}+49 n^{2}+1} $
$b_6=\frac{n \left(n^{108}+72 n^{106}+6234 n^{102}-31104 n^{100}-36 n^{98}+569433 n^{96}+3063744 n^{94}+15552 n^{92}-1574800 n^{90}+64105344 n^{88}-1531872 n^{86}+165039924 n^{84}+242797536 n^{82}-32052672 n^{80}+963577944 n^{78}-594241920 n^{76}-121398768 n^{74}+4018389060 n^{72}+20450880 n^{70}+297120960 n^{68}+8815738128 n^{66}-329795712 n^{64}-10225440 n^{62}+12135907374 n^{60}+1187302320 n^{58}+164897856 n^{56}+16524170140 n^{54}-329795712 n^{52}-593651160 n^{50}+12135907374 n^{48}+20450880 n^{46}+164897856 n^{44}+8815738128 n^{42}-594241920 n^{40}-10225440 n^{38}+4018389060 n^{36}+242797536 n^{34}+297120960 n^{32}+963577944 n^{30}+64105344 n^{28}-121398768 n^{26}+165039924 n^{24}+3063744 n^{22}-32052672 n^{20}-1574800 n^{18}-31104 n^{16}-1531872 n^{14}+569433 n^{12}+72 n^{10}+15552 n^{8}+6234 n^{6}-36 n^{2}+1\right)}{n^{108}-36 n^{106}+6234 n^{102}+15552 n^{100}+72 n^{98}+569433 n^{96}-1531872 n^{94}-31104 n^{92}-1574800 n^{90}-32052672 n^{88}+3063744 n^{86}+165039924 n^{84}-121398768 n^{82}+64105344 n^{80}+963577944 n^{78}+297120960 n^{76}+242797536 n^{74}+4018389060 n^{72}-10225440 n^{70}-594241920 n^{68}+8815738128 n^{66}+164897856 n^{64}+20450880 n^{62}+12135907374 n^{60}-593651160 n^{58}-329795712 n^{56}+16524170140 n^{54}+164897856 n^{52}+1187302320 n^{50}+12135907374 n^{48}-10225440 n^{46}-329795712 n^{44}+8815738128 n^{42}+297120960 n^{40}+20450880 n^{38}+4018389060 n^{36}-121398768 n^{34}-594241920 n^{32}+963577944 n^{30}-32052672 n^{28}+242797536 n^{26}+165039924 n^{24}-1531872 n^{22}+64105344 n^{20}-1574800 n^{18}+15552 n^{16}+3063744 n^{14}+569433 n^{12}-36 n^{10}-31104 n^{8}+6234 n^{6}+72 n^{2}+1} $
$b_7=\frac{n \left(n^{150}+97 n^{148}-98 n^{146}+15705 n^{144}-123784 n^{142}+108128 n^{140}+3446348 n^{138}+24162948 n^{136}-27663384 n^{134}-54131940 n^{132}+1284695208 n^{130}-1216696704 n^{128}+6634254058 n^{126}-229454414 n^{124}-5801634356 n^{122}+81110865930 n^{120}-223888114072 n^{118}+145342483424 n^{116}+785626162028 n^{114}-1023645380268 n^{112}+167080903272 n^{110}+4270749389916 n^{108}-2214069073416 n^{106}-2065478795328 n^{104}+11467933754407 n^{102}+4703023132063 n^{100}-14705536010894 n^{98}+29106714141151 n^{96}+16623299156016 n^{94}-36970181795136 n^{92}+54069311943896 n^{90}+39190788849032 n^{88}-74153669206960 n^{86}+81131871039992 n^{84}+53330412573456 n^{82}-96765223483392 n^{80}+100555075829164 n^{78}+64281686810620 n^{76}-117821240784152 n^{74}+100555075829164 n^{72}+52483881399888 n^{70}-95918692309824 n^{68}+81131871039992 n^{66}+35216158023752 n^{64}-70179038381680 n^{62}+54069311943896 n^{60}+16621068832368 n^{58}-36967951471488 n^{56}+29106714141151 n^{54}+3131577515695 n^{52}-13134090394526 n^{50}+11467933754407 n^{48}-265305036072 n^{46}-4014242832672 n^{44}+4270749389916 n^{42}-307766370060 n^{40}-548798106936 n^{38}+785626162028 n^{36}-77870021944 n^{34}-675608704 n^{32}+81110865930 n^{30}+3907562866 n^{28}-9938651636 n^{26}+6634254058 n^{24}+623897544 n^{22}-555899040 n^{20}-54131940 n^{18}+13727076 n^{16}-17227512 n^{14}+3446348 n^{12}-53992 n^{10}+38336 n^{8}+15705 n^{6}+49 n^{4}-50 n^{2}+1\right)}{n^{150}-50 n^{148}+49 n^{146}+15705 n^{144}+38336 n^{142}-53992 n^{140}+3446348 n^{138}-17227512 n^{136}+13727076 n^{134}-54131940 n^{132}-555899040 n^{130}+623897544 n^{128}+6634254058 n^{126}-9938651636 n^{124}+3907562866 n^{122}+81110865930 n^{120}-675608704 n^{118}-77870021944 n^{116}+785626162028 n^{114}-548798106936 n^{112}-307766370060 n^{110}+4270749389916 n^{108}-4014242832672 n^{106}-265305036072 n^{104}+11467933754407 n^{102}-13134090394526 n^{100}+3131577515695 n^{98}+29106714141151 n^{96}-36967951471488 n^{94}+16621068832368 n^{92}+54069311943896 n^{90}-70179038381680 n^{88}+35216158023752 n^{86}+81131871039992 n^{84}-95918692309824 n^{82}+52483881399888 n^{80}+100555075829164 n^{78}-117821240784152 n^{76}+64281686810620 n^{74}+100555075829164 n^{72}-96765223483392 n^{70}+53330412573456 n^{68}+81131871039992 n^{66}-74153669206960 n^{64}+39190788849032 n^{62}+54069311943896 n^{60}-36970181795136 n^{58}+16623299156016 n^{56}+29106714141151 n^{54}-14705536010894 n^{52}+4703023132063 n^{50}+11467933754407 n^{48}-2065478795328 n^{46}-2214069073416 n^{44}+4270749389916 n^{42}+167080903272 n^{40}-1023645380268 n^{38}+785626162028 n^{36}+145342483424 n^{34}-223888114072 n^{32}+81110865930 n^{30}-5801634356 n^{28}-229454414 n^{26}+6634254058 n^{24}-1216696704 n^{22}+1284695208 n^{20}-54131940 n^{18}-27663384 n^{16}+24162948 n^{14}+3446348 n^{12}+108128 n^{10}-123784 n^{8}+15705 n^{6}-98 n^{4}+97 n^{2}+1} $
and so on.
Note 1: In the first family, I think it is nice to denote $b_1=\frac{n\times \color{red}{1}}{1}.$
For the first family, the sum of the coefficients of $b_m$'s denominator is $2^{k(m)}$ where $k(m)=0, 4, 8, 16, 24, 36, 48, 64, 80, 100, 120, 144, 168, 196, 224, 256.$ (See https://oeis.org/A137932).
For the second family, the sum of the coefficients of $b_m$'s denominator is $2^{k(m)}$ where $k(m)= ?$
I have verified the first for $1\leq m \leq 16 $.
Note 2:
For the first family, the degree of $b_m$'s denominator is $d(m)=2\alpha(m)$ where $\alpha(m)=0, 3, 4, 9, 12, 19, 24, 33, 40, 51, 60, 73, 84, 99, 112.$ (See https://oeis.org/A097063).
For the second family, the degree of $b_m$'s denominator is $d(m)=6\beta(m)$ where $\beta(m)=1, 2, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98, 113.$ (See https://oeis.org/A000982).
It is worth noting,
$$\alpha(m)-\beta(m)=(-1)^m$$
I have verified this for $1\leq m \leq 15 $. I think it is true for all $m\geq 1$.
