Computed 78519 primes, largest is 1000253
Extra primes in list: 
Testing some points:
The points are P0 = [0:2:1], P1 = [1:0:1], and P2 = [2:0:1]
Now in Pari format:
The points are P0 = [0,2], P1 = [1,0], and P2 = [2,0]
Their negatives are -P0 = [0:-3:1], -P1 = [1:-1:1], and -P2 = [2:-1:1]
Computing their heights:
Heights are 0.99090633315308797388259855289, 0.66820516565192793503314205089, and 0.76704335533154620579545064655
The origin is [0:1:0]
Now some additions etc,:
P0 + P1 = [3:3:1]
P0 - P1 = [8:21:1]
P0 - P1 - P2 = [93:-143:27]
P0.twice() = [245:-32:125]
P0 + P0 = [245:-32:125]
3*P0 = [-74725:-438957:117649]
P0 - P0 = [0:1:0]
P0 +3 P1 - P2 = [816:-23310:1]
2P0 +2 P1 + P2 = [93:896:1]
Now we try a systematic exploration
-3, -3, -3: [173402493:-312412055:41781923]
-3, -3, -2: [4189408:8960581:704969]
-3, -3, -1: [1786410:-1787743:729000]
-3, -3, 0: [1735612:8628750:68921]
-3, -3, 1: [160962987:-73608612:81182737]
-3, -3, 2: [1356164797:-71126160883:493039]
-3, -3, 3: [125658707472212:6072729019689:62179146498113]
-3, -2, -3: [2212614:-10179939:4574296]
-3, -2, -2: [-34216:251158:103823]
-3, -2, -1: [-17639:-109980:29791]
-3, -2, 0: [92379:143623:148877]
-3, -2, 1: [-4163000:-7596119:1953125]
-3, -2, 2: [1838781714:-135929375:1806932232]
-3, -2, 3: [-4773144746116:-976602707934:1584834464179]
-3, -1, -3: [1178299:-4850852:68921]
-3, -1, -2: [94875:71299:35937]
-3, -1, -1: [24412:-50127:4913]
-3, -1, 0: [85098:154071:17576]
-3, -1, 1: [3858272:-4442750:1442897]
-3, -1, 2: [587381677:2315829684:36264691]
-3, -1, 3: [5308466179635:-3022905914377:2608126472125]
-3, 0, -3: [4430992:-3892417:4330747]
-3, 0, -2: [-119434:155991:54872]
-3, 0, -1: [32412:-97546:50653]
-3, 0, 0: [-74725:321308:117649]
-3, 0, 1: [-1453671:-16871239:5000211]
-3, 0, 2: [1175800444:3238879023:2554497863]
-3, 0, 3: [-8489731927494:-19600265861445:4763680807976]
-3, 1, -3: [5081640:197180774:3375]
-3, 1, -2: [5310773:-336300:2685619]
-3, 1, -1: [657459:-3409809:24389]
-3, 1, 0: [8348488:4750403:3442951]
-3, 1, 1: [228191806:-536379633:37259704]
-3, 1, 2: [114813702300:174977582986:28288984823]
-3, 1, 3: [556368785287493:-733074121389148:187375095748567]
-3, 2, -3: [8704995703:942557295:8960030533]
-3, 2, -2: [-363511764:-185058623:122763473]
-3, 2, -1: [216346510:-146306363:205379000]
-3, 2, 0: [-724812816:692674750:291434247]
-3, 2, 1: [28678925057:-61748215932:37360194607]
-3, 2, 2: [-8685736238109:26728697273751:8963913629917]
-3, 2, 3: [-1258675911627328:-170609866818516791:56286247430493053]
-3, 3, -3: [6411102385354:21118267149921:548548738856]
-3, 3, -2: [734465224452:-495548440926:348765000319]
-3, 3, -1: [46328051291:614238485020:263374721]
-3, 3, 0: [2041384645877:-367524023451:1045987472951]
-3, 3, 1: [22947655893900:-160349260775081:470003609375]
-3, 3, 2: [49052364371736622:19768548114717625:21681696304639736]
-3, 3, 3: [141154686087724689336:-382997118708972871786:18067866971533021791]
-2, -3, -3: [9718082:-5196113:9129329]
-2, -3, -2: [-88288:60819:32768]
-2, -3, -1: [10350:-17696:12167]
-2, -3, 0: [-8569:21344:6859]
-2, -3, 1: [22083:-365579:132651]
-2, -3, 2: [1467346:29518545:15813251]
-2, -3, 3: [-14840686668:-53202497391:13077534016]
-2, -2, -3: [55278:480700:729]
-2, -2, -2: [4277:-1152:2197]
-2, -2, -1: [93:-897:1]
-2, -2, 0: [742:209:343]
-2, -2, 1: [5000:-15351:512]
-2, -2, 2: [1124214:1218856:357911]
-2, -2, 3: [1399931107:-2331158336:371694959]
-2, -1, -3: [20213:12411:24389]
-2, -1, -2: [-330:-377:125]
-2, -1, -1: [68:-25:64]
-2, -1, 0: [-78:28:27]
-2, -1, 1: [1243:-1632:1331]
-2, -1, 2: [-128097:250701:79507]
-2, -1, 3: [83553550:-541263919:223648543]
-2, 0, -3: [28688:68991:4096]
-2, 0, -2: [798:-720:343]
-2, 0, -1: [37:224:1]
-2, 0, 0: [245:-93:125]
-2, 0, 1: [342:-6325:1]
-2, 0, 2: [650692:91633:314432]
-2, 0, 3: [448295118:-1637295092:33076161]
-2, 1, -3: [-19875:324544:148877]
-2, 1, -2: [-1807:-8533:2197]
-2, 1, -1: [522:559:729]
-2, 1, 0: [-1208:-1845:512]
-2, 1, 1: [31062:-6216:29791]
-2, 1, 2: [-4555265:134912:1520875]
-2, 1, 3: [9633434827:-9849844875:9677214091]
-2, 2, -3: [22319442:19888197:7880599]
-2, 2, -2: [371756:-695903:85184]
-2, 2, -1: [67850:138204:12167]
-2, 2, 0: [199563:-208288:79507]
-2, 2, 1: [1507693:6967591:68921]
-2, 2, 2: [936156018:-458727065:469097433]
-2, 2, 3: [13269197888:-2985362435769:262144]
-2, 3, -3: [-4964538862:7806531200:2565726409]
-2, 3, -2: [67004241:-268411616:125751501]
-2, 3, -1: [-8104163:48961227:19465109]
-2, 3, 0: [-21256194:-153135513:42508549]
-2, 3, 1: [836123500:1525038911:1450571968]
-2, 3, 2: [-290048080358:-567945909132:142809632083]
-2, 3, 3: [924932335876083:-16318887357184:920911723813171]
-1, -3, -3: [11454454:20427225:2406104]
-1, -3, -2: [80352:-93666:29791]
-1, -3, -1: [1955:7564:125]
-1, -3, 0: [2717:-1575:1331]
-1, -3, 1: [816:23309:1]
-1, -3, 2: [798682:-68303:405224]
-1, -3, 3: [119498676:-651053998:4019679]
-1, -2, -3: [-33633:139628:50653]
-1, -2, -2: [-91:-1147:343]
-1, -2, -1: [12:35:27]
-1, -2, 0: [-14:-33:8]
-1, -2, 1: [120:18:125]
-1, -2, 2: [-6461:-3700:2197]
-1, -2, 3: [4922993:-3160443:4657463]
-1, -1, -3: [11832:6549:4913]
-1, -1, -2: [50:-119:8]
-1, -1, -1: [4:6:1]
-1, -1, 0: [3:-4:1]
-1, -1, 1: [11:35:1]
-1, -1, 2: [1548:-1079:729]
-1, -1, 3: [139610:1648791:1000]
-1, 0, -3: [-3344:3094:1331]
-1, 0, -2: [21:-44:27]
-1, 0, -1: [-1:3:1]
-1, 0, 0: [0:-3:1]
-1, 0, 1: [2:13:8]
-1, 0, 2: [-476:-1422:343]
-1, 0, 3: [506217:208180:571787]
-1, 1, -3: [30475:-5797:15625]
-1, 1, -2: [52:-375:1]
-1, 1, -1: [18:7:8]
-1, 1, 0: [8:-22:1]
-1, 1, 1: [93:116:27]
-1, 1, 2: [7475:-11361:2197]
-1, 1, 3: [4040376:10769499:493039]
-1, 2, -3: [-440586:-390925:157464]
-1, 2, -2: [5236:-2538:4913]
-1, 2, -1: [-345:204:125]
-1, 2, 0: [301:-475:343]
-1, 2, 1: [-1804:4179:1331]
-1, 2, 2: [45066:-518089:195112]
-1, 2, 3: [3497024:311754698:158340421]
-1, 3, -3: [193462973:-156569868:86938307]
-1, 3, -2: [409317:3155489:6859]
-1, 3, -1: [202288:-61085:103823]
-1, 3, 0: [27222:-305641:216]
-1, 3, 1: [1828180:436722:857375]
-1, 3, 2: [84254013:-271587900:7880599]
-1, 3, 3: [144166484837:146955288613:47555965367]
0, -3, -3: [-198562138:65431412:68417929]
0, -3, -2: [499689:-641312:531441]
0, -3, -1: [-8075:15471:4913]
0, -3, 0: [858:-5253:2197]
0, -3, 1: [-752:9191:4096]
0, -3, 2: [-60458:-305040:79507]
0, -3, 3: [22469403:26544416:32461759]
0, -2, -3: [2021717:-787325:1030301]
0, -2, -2: [406:-8181:1]
0, -2, -1: [132:17:64]
0, -2, 0: [14:-52:1]
0, -2, 1: [75:64:27]
0, -2, 2: [1547:-2967:343]
0, -2, 3: [369738:730191:68921]
0, -1, -3: [-32232:-50399:13824]
0, -1, -2: [130:-24:125]
0, -1, -1: [-3:0:1]
0, -1, 0: [1:-1:1]
0, -1, 1: [-2:3:1]
0, -1, 2: [36:-133:64]
0, -1, 3: [-5750:31324:12167]
0, 0, -3: [17347:-18336:6859]
0, 0, -2: [21:95:1]
0, 0, -1: [2:-1:1]
0, 0, 0: [0:1:0]
0, 0, 1: [2:0:1]
0, 0, 2: [21:-96:1]
0, 0, 3: [17347:11477:6859]
0, 1, -3: [-5750:-43491:12167]
0, 1, -2: [36:69:64]
0, 1, -1: [-2:-4:1]
0, 1, 0: [1:0:1]
0, 1, 1: [-3:-1:1]
0, 1, 2: [130:-101:125]
0, 1, 3: [-32232:36575:13824]
0, 2, -3: [369738:-799112:68921]
0, 2, -2: [1547:2624:343]
0, 2, -1: [75:-91:27]
0, 2, 0: [14:51:1]
0, 2, 1: [132:-81:64]
0, 2, 2: [406:8180:1]
0, 2, 3: [2021717:-242976:1030301]
0, 3, -3: [22469403:-59006175:32461759]
0, 3, -2: [-60458:225533:79507]
0, 3, -1: [-752:-13287:4096]
0, 3, 0: [858:3056:2197]
0, 3, 1: [-8075:-20384:4913]
0, 3, 2: [499689:109871:531441]
0, 3, 3: [-198562138:-133849341:68417929]
1, -3, -3: [144166484837:-194511253980:47555965367]
1, -3, -2: [84254013:263707301:7880599]
1, -3, -1: [1828180:-1294097:857375]
1, -3, 0: [27222:305425:216]
1, -3, 1: [202288:-42738:103823]
1, -3, 2: [409317:-3162348:6859]
1, -3, 3: [193462973:69631561:86938307]
1, -2, -3: [3497024:-470095119:158340421]
1, -2, -2: [45066:322977:195112]
1, -2, -1: [-1804:-5510:1331]
1, -2, 0: [301:132:343]
1, -2, 1: [-345:-329:125]
1, -2, 2: [5236:-2375:4913]
1, -2, 3: [-440586:233461:157464]
1, -1, -3: [4040376:-11262538:493039]
1, -1, -2: [7475:9164:2197]
1, -1, -1: [93:-143:27]
1, -1, 0: [8:21:1]
1, -1, 1: [18:-15:8]
1, -1, 2: [52:374:1]
1, -1, 3: [30475:-9828:15625]
1, 0, -3: [506217:-779967:571787]
1, 0, -2: [-476:1079:343]
1, 0, -1: [2:-21:8]
1, 0, 0: [0:2:1]
1, 0, 1: [-1:-4:1]
1, 0, 2: [21:17:27]
1, 0, 3: [-3344:-4425:1331]
1, 1, -3: [139610:-1649791:1000]
1, 1, -2: [1548:350:729]
1, 1, -1: [11:-36:1]
1, 1, 0: [3:3:1]
1, 1, 1: [4:-7:1]
1, 1, 2: [50:111:8]
1, 1, 3: [11832:-11462:4913]
1, 2, -3: [4922993:-1497020:4657463]
1, 2, -2: [-6461:1503:2197]
1, 2, -1: [120:-143:125]
1, 2, 0: [-14:25:8]
1, 2, 1: [12:-62:27]
1, 2, 2: [-91:804:343]
1, 2, 3: [-33633:-190281:50653]
1, 3, -3: [119498676:647034319:4019679]
1, 3, -2: [798682:-336921:405224]
1, 3, -1: [816:-23310:1]
1, 3, 0: [2717:244:1331]
1, 3, 1: [1955:-7689:125]
1, 3, 2: [80352:63875:29791]
1, 3, 3: [11454454:-22833329:2406104]
2, -3, -3: [924932335876083:-904592836455987:920911723813171]
2, -3, -2: [-290048080358:425136277049:142809632083]
2, -3, -1: [836123500:-2975610879:1450571968]
2, -3, 0: [-21256194:110626964:42508549]
2, -3, 1: [-8104163:-68426336:19465109]
2, -3, 2: [67004241:142660115:125751501]
2, -3, 3: [-4964538862:-10372257609:2565726409]
2, -2, -3: [13269197888:2985362173625:262144]
2, -2, -2: [936156018:-10370368:469097433]
2, -2, -1: [1507693:-7036512:68921]
2, -2, 0: [199563:128781:79507]
2, -2, 1: [67850:-150371:12167]
2, -2, 2: [371756:610719:85184]
2, -2, 3: [22319442:-27768796:7880599]
2, -1, -3: [9633434827:172630784:9677214091]
2, -1, -2: [-4555265:-1655787:1520875]
2, -1, -1: [31062:-23575:29791]
2, -1, 0: [-1208:1333:512]
2, -1, 1: [522:-1288:729]
2, -1, 2: [-1807:6336:2197]
2, -1, 3: [-19875:-473421:148877]
2, 0, -3: [448295118:1604218931:33076161]
2, 0, -2: [650692:-406065:314432]
2, 0, -1: [342:6324:1]
2, 0, 0: [245:-32:125]
2, 0, 1: [37:-225:1]
2, 0, 2: [798:377:343]
2, 0, 3: [28688:-73087:4096]
2, 1, -3: [83553550:317615376:223648543]
2, 1, -2: [-128097:-330208:79507]
2, 1, -1: [1243:301:1331]
2, 1, 0: [-78:-55:27]
2, 1, 1: [68:-39:64]
2, 1, 2: [-330:252:125]
2, 1, 3: [20213:-36800:24389]
2, 2, -3: [1399931107:1959463377:371694959]
2, 2, -2: [1124214:-1576767:357911]
2, 2, -1: [5000:14839:512]
2, 2, 0: [742:-552:343]
2, 2, 1: [93:896:1]
2, 2, 2: [4277:-1045:2197]
2, 2, 3: [55278:-481429:729]
2, 3, -3: [-14840686668:40124963375:13077534016]
2, 3, -2: [1467346:-45331796:15813251]
2, 3, -1: [22083:232928:132651]
2, 3, 0: [-8569:-28203:6859]
2, 3, 1: [10350:5529:12167]
2, 3, 2: [-88288:-93587:32768]
2, 3, 3: [9718082:-3933216:9129329]
3, -3, -3: [141154686087724689336:364929251737439849995:18067866971533021791]
3, -3, -2: [49052364371736622:-41450244419357361:21681696304639736]
3, -3, -1: [22947655893900:159879257165706:470003609375]
3, -3, 0: [2041384645877:-678463449500:1045987472951]
3, -3, 1: [46328051291:-614501859741:263374721]
3, -3, 2: [734465224452:146783440607:348765000319]
3, -3, 3: [6411102385354:-21666815888777:548548738856]
3, -2, -3: [-1258675911627328:114323619388023738:56286247430493053]
3, -2, -2: [-8685736238109:-35692610903668:8963913629917]
3, -2, -1: [28678925057:24388021325:37360194607]
3, -2, 0: [-724812816:-984108997:291434247]
3, -2, 1: [216346510:-59072637:205379000]
3, -2, 2: [-363511764:62295150:122763473]
3, -2, 3: [8704995703:-9902587828:8960030533]
3, -1, -3: [556368785287493:545699025640581:187375095748567]
3, -1, -2: [114813702300:-203266567809:28288984823]
3, -1, -1: [228191806:499119929:37259704]
3, -1, 0: [8348488:-8193354:3442951]
3, -1, 1: [657459:3385420:24389]
3, -1, 2: [5310773:-2349319:2685619]
3, -1, 3: [5081640:-197184149:3375]
3, 0, -3: [-8489731927494:14836585053469:4763680807976]
3, 0, -2: [1175800444:-5793376886:2554497863]
3, 0, -1: [-1453671:11871028:5000211]
3, 0, 0: [-74725:-438957:117649]
3, 0, 1: [32412:46893:50653]
3, 0, 2: [-119434:-210863:54872]
3, 0, 3: [4430992:-438330:4330747]
3, 1, -3: [5308466179635:414779442252:2608126472125]
3, 1, -2: [587381677:-2352094375:36264691]
3, 1, -1: [3858272:2999853:1442897]
3, 1, 0: [85098:-171647:17576]
3, 1, 1: [24412:45214:4913]
3, 1, 2: [94875:-107236:35937]
3, 1, 3: [1178299:4781931:68921]
3, 2, -3: [-4773144746116:-608231756245:1584834464179]
3, 2, -2: [1838781714:-1671002857:1806932232]
3, 2, -1: [-4163000:5642994:1953125]
3, 2, 0: [92379:-292500:148877]
3, 2, 1: [-17639:80189:29791]
3, 2, 2: [-34216:-354981:103823]
3, 2, 3: [2212614:5605643:4574296]
3, 3, -3: [125658707472212:-68251875517802:62179146498113]
3, 3, -2: [1356164797:71125667844:493039]
3, 3, -1: [160962987:-7574125:81182737]
3, 3, 0: [1735612:-8697671:68921]
3, 3, 1: [1786410:1058743:729000]
3, 3, 2: [4189408:-9665550:704969]
3, 3, 3: [173402493:270630132:41781923]
P0 -P1 -P2 = [93:-143:27]
Height of     P0-P1-P2 = 3.518883255539141483787954523
3 (P0 -P1 -P2) = [141154686087724689336:364929251737439849995:18067866971533021791]
Height of 3*(P0-P1-P2) = 31.669949299852273354091590707
The quotient is 9