curve #4232

y² = x³ + x² − 6040838479798342853233054719334955442838625x + 5615244060673613371079850974429551701050429956355245459699359375

a-invariants: [0, 1, 0, -6040838479798342853233054719334955442838625, 5615244060673613371079850974429551701050429956355245459699359375]
c-invariants: [289960247030320456955186626528077861256254016, -4851570868422001952614731003389314592449302602050100544347784024064]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 6434609419411114392518370233005172295466214234133422996628607416819300269168594580294781406755228672665120
discriminant (Δ): 486823963672333392582867722630903457554234289416613310036927824287345560159549127784869155025080378381995437663536649330880000000
Faltings height: 23.4844
naive height: 307.1350
primes of bad reduction: 2, 3, 5, 7, 17, 19, 79, 191, 277, 9720779799103543067, 145928143603050514603893653574754050024972430623083940865898646919490123429
regulator: 1194660859165830912894171500910.8895895679015664839914926802109206481527119302399102809765679
submitted by: apkorte
submitted at: 2026-10-14 23:42:05 UTC
last updated: 2026-10-14 23:42:05 UTC

Witness: 27 independent points log in to add more points →

(1641948627672048959875, -11098801066633942193052538036500)
(-493982288267120708750, -92080239239550754075615072297875)
(-2804099133531837941045, 22490023873082274284852145380220)
(961963751818169954425, -26350576982753697774019177647300)
(7183051418661664787125/4, -189017252504944422350843585223375/8)
(1880450068746205848075, -30086446076630801348622186584700)
(1122588870462862733125, 15765753684052549560956379471000)
(2583245238136933331125, 85139151871027424643204531714000)
(-967744635235413534395, 102737099040244763936208152584680)
(1068029087943251200675, 19538137542038086861193419262700)
(-695371959924938861195, 97363402223899329628990194141120)
(63811821465942547083445, 16107718422862235419827033852438480)
(14816232354340530422875, 1780051427744822153211571236490500)
(-2824604270211913320125, -11934395139766110473016899956500)
(5458253070675884498875, 367774344655350107458139431429500)
(1783172295059813379865, 22657169160058640501227388696460)
(4598020887139317531749845/2209, 4711773124774495437361435257691984560/103823)
(6401103820755014754528625/2809, 9034852978019857016391095335443763500/148877)
(-2384912079909926545955/4, 759157824264928344569224237536315/8)
(1730927467374931027525, 18575244992951851369769632194600)
(9852741903783309333417625/6241, 1789514644040462801230673321979670500/493039)
(15827511372376590551065, -1968490670043892850056764394837260)
(2667192352016921798725, -92072502204840295547843392223400)
(319441183260748427883325/169, 67722902780713893546800011770429600/2197)
(51313647951794919063931/25, -5395901567208689939221012281755004/125)
(-257180489199835982859874175/94249, 1221072834130143777782825572068980903100/28934443)
(-1907599197648762576125, -100980759484179275186143134004500)

Commentary

Specialization at T = 5934/4651 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -76081088435681081513047T^8 + 352756680332626657087252T^7 - 82339513624259414646469T^6 - 901545658338022693466490T^5 + 229450949582480650416915T^4 - 1247200030321945517143512T^3 - 1146369340545356004506811T^2 + 1687352936729813079668190T - 411496468647819081255900, A6 = 8112841022538000650593845586374970T^12 - 57168146144925559393934264903441092T^11 + 74672050992321347131912066302099019T^10 + 277662613390589143744390101854640120T^9 - 608279964343956992513678172184166364T^8 - 775627450421715998790683046858185472T^7 + 1913663946413950754123492444913586362T^6 + 1711261933991806885632762747792403176T^5 - 2007957630504188399500442532556750554T^4 - 505502209838775107774478816852469020T^3 + 1225508258092292403296873724948672675T^2 - 576830173318718574016743233507280000T + 96363113602603152516124313597692500. It is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-14 23:42:05 UTC · history