curve #542

y² = x³ + x² − 54113848533568140361225153191279559014768945x + 147661503586524465474159358005052226244960662130382591800920369043

a-invariants: [0, 1, 0, -54113848533568140361225153191279559014768945, 147661503586524465474159358005052226244960662130382591800920369043]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 13858119093380870911402836699917707730917559748643075892856552587147896304378790356681414169480 ★ record for rank ≥ 26
discriminant (Δ): 722277800241862558247745792286000446849914177976842506688452357308161888108929747302648556648102151776258683140249105325544020000000
Faltings height: 24.0667
naive height: 313.7126
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 31, 41, 59, 71, 107, 109, 127, 34634663044790853665781117256440766636432529436532867313481217347957006763
regulator: 319376359427026699537144218.811704398751567924054406595344856994726103082917876605303291831
submitted by: wgxli
submitted at: 2026-09-02 22:55:57 UTC
last updated: 2026-09-02 22:55:57 UTC

Witness: 26 independent points log in to add more points →

(463258392246256108402548539/16129, -9670428945943007908612524840155737452090/2048383)
(618346097591468452406422299/126025, -735623789789511244635378943525989929118/44738875)
(29243161090096769780811670292145369/10834182907024, -5200196743287718422965396541881952356399993412190875/35661059732322520768)
(1086564072638274496220516344814621457/3888456654889, 1132228128779567049165879938364310240724310525915169866/7667713781538752213)
(9577034980207489886273847660947679/2683872892009, 2662948257154870023417046927989572010618580837350/4396862816952420277)
(1944590826107820969183528801/724201, 90822747450981073444426530167193732585690/616295051)
(2273335591534512359709313491743406608451/747683678873335681, 68532011847466007928314778714678043662215428012882595912070/646512387888978843803609921)
(70341205449114774654485352202746331/14078997335601, -2369990593776010637734976231185477886374832535824650/52827199723644736599)
(-24903181690476962243687312181/35485849, 90993606704516611026770496637554337786127250/211389202493)
(536715243879219071964844316369464488561/39684966343517881, -10867192601129420288755330982592210811533808129541917125750/7905676234678286155869779)
(-89879781952122042540868190825409/2131071512761, -1204650835117111998142186335413587895103377389197750/3110978684687250259)
(161338359469864372784769/49, 24690299037581828526680868054609750/343)
(61947683284548929115971646/5041, 414071172590296198816402280795372072625/357911)
(1208012293120536393669555728376488722269039/28242484780909296769, 1309209453614681724186055037123711917141414023447341030451881550/150090897683781035590332919103)
(8146235081033964942814631901/2515396, 318596559655511591726893151862736743142875/3989418056)
(325240380464434950102147081262804679/92599299859609, 27105738931596700323178626901038461104764500639083050/891069450451938044477)
(1901275846139021277246056622798046443/216772883257681, -1882103370194226439720653141194559299033595924675027498/3191592465735438218329)
(760950591440087347814120498199/138886225, -204736986965344611166480954583422604222584482/1636774161625)
(21769718603641580521091981908803376405609/2520876527982746649, -2279553241511384059227102461525136314378335064194804590898610/4002463331366847495485599107)
(27726587581638663512504657091938658496779/5658571247560374649, -165592686322256639098183497430571699226485082569118635565390/13460468707811116072653794893)
(208495835569894820274601826984401854/42556665461209, 3276105967670823309610460502238777141082068287139875/277620407299473588323)
(19375015165575160397384026307244011589980723451/2103290416171621509192721, -2002235552720523505496620536826128796526379903883309199960935626513450/3050344316738427953253214471280635769)
(22899598222771050507806736069690542548731144081/4538378219933617255532176, -536541121295189239650939635084114684923197336621762890318249344641375/9668320297373739172628243027644191424)
(25265952281542283246499666647405421099/3321512317948969, -80349084239714038184444076812931321698662860981153998910/191427267919469986677547)
(150940712012002841731922418161269417471808809/29659343948684348771889, -326029472639582640861959778804984802302098626574531964756957074750/5107899110061461374331207906970537)
(2152516194917313372124970158676364129/396516895703824, -930120684248276603035968078887018477591121864449961465/7895734677622198687168)

Commentary

Specialization at T = 1889/8 of the rank-16 fibration of X948 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 38104T^2 + 695028T + 78468859, p = 72T^2 + 537067T + 2394620, q = -3786T^2 + 278773T + 146946125, D = 900018198T^4 - 74161080925T^3 - 2169781853671T^2 + 80721528900062T + 339935259286920, E = 65158368T^4 - 55322669902T^3 - 73322018893T^2 + 82814330071091T - 68006574064560.

last edited by wgxli at 2026-09-17 17:33:59 UTC · history