curve #4204

y² + xy + y = x³ − x² − 1621238076635217389723521337019699459918297x + 796788001285152932898167954720217974205576571662448287559822521

a-invariants: [1, -1, 1, -1621238076635217389723521337019699459918297, 796788001285152932898167954720217974205576571662448287559822521]
c-invariants: [77819427678490434706729024176945574076078241, -688424833110372134023666925453715122757437877307559065368344306289]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 2967355291463937048383780024918256242192132464440126497308674951954423661079347962552287794552076810
discriminant (Δ): -1542206512749950257160613497475126110089398017151089574666554577062896021963744582072395315009675056967342713184844513280000000
Faltings height: 23.0789
naive height: 303.1945
primes of bad reduction: 2, 5, 7, 11, 17, 19, 23, 37, 89, 21481, 5182753318937, 1414951979967742058549850114727406900636749336334406032440944959867012917
regulator: 11523870156431655305799520657.5457908049794392665106363984524610965928710173094582609543325
submitted by: apkorte
submitted at: 2026-10-13 03:41:19 UTC
last updated: 2026-10-13 03:41:19 UTC

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

(-1440789547016287468889, -11905988733267710368254710441556)
(765760199599265544261, 2083632232564070894563948124394)
(975933547599719656871, 12003818545931977567020299481004)
(4572794775960877378311, -298332708485244129025386323707956)
(473399742844765051111, 11635577283624067175168138830444)
(729531088583804123911, 1520610118111001832546071504844)
(-1546225711825448788001/9, -883296391283205333372019555042012/27)
(-86558494715533493849, 30601820211345001685061324334764)
(770920963752770046111, 2261561200093985862977747850444)
(-901540235052476125609, -39059572166491764933461061876676)
(806781503045277844231, 3732864827539558589899132911564)
(-1414552313080694895289, -16113781577856248027144028479956)
(46748149417492366326679/169, -42231169188762851896564375461910652/2197)
(866652304409985184679, -6532142583045928164622829238356)
(-164476027387783990003601/2809, -4445208899243012403994559063645023812/148877)
(772775784140916732071, -2328643912905723847198220168276)
(1276412917294981283111, -28407550283856303098297314345556)
(1118275965621592787751, 19551125830090584861924565712684)
(-1201764831556674119609, 31772623670590533956019947809324)
(894368653327463393611, -7886999274077429897151442541556)
(1355020626118526286985471/2025, -309790344293392174631976992703318116/91125)
(783352704491052676711, -2735775269082494490428214588756)
(205568702699846001975529/289, 9166235107510577102146735556686022/4913)
(494124134865546019411, -10786094113668252616544478291156)
(7641809246567536965617791/11881, 5806746306245217647173319896360734876/1295029)
(447902079597334380711, -12668407203745807444570670236756)

Commentary

Specialization at T = -318/133 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = 1840758517133685517328T^8 - 28551969868039231295840T^7 - 1772998693252999313402080T^6 - 759387847760867496814400T^5 - 598922555435041558626640T^4 - 544800986643777256848080T^3 + 926816232306216461908040T^2 + 288493294443742559437240T - 230970134041531372322875, A6 = 325113825546664597334681997252736T^12 + 13439521326531181915894800771978880T^11 + 138327069120450496577609764257248960T^10 + 3518007078129231511599849398766400T^9 + 1769321142185421847687389290390981520T^8 + 1594947821477300217886357939071742368T^7 - 1479100347162091017597427825166729760T^6 - 1235366662137150944785202891205624960T^5 + 731032719531350653488345005675279040T^4 + 467848968404338281084037609996168720T^3 - 249593152301734103394166433543833544T^2 - 84246673758026015337923892531146920T + 42769382258496484819391456728787350. 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 2026-10-13 03:41:19 UTC · history