curve #4576

y² + xy + y = x³ − x² − 8724712962489323569656597636958905707541377x + 9916041896027202147574972224790470738820335636667528796798682001

a-invariants

[1, -1, 1, -8724712962489323569656597636958905707541377, 9916041896027202147574972224790470738820335636667528796798682001]

c-invariants

[418786222199487531343516686574027473961986081, -8567460198167502655502891464219069024449724164991161756801232311729]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

34984200010319809978334193632379682476220287211750140365628784855559193613974153895640025232509783596107677770490

discriminant (Δ)

26706629531322531858945378632699327672220905127683241133082203465330267181688709593638973901891276758922495402071134763456000000

Faltings height

23.4193

naive height

308.2378

primes of bad reduction

2, 3, 5, 17, 29, 103, 5431, 836654506425293460683756935665429791599, 1684687433819774755654444078936826242963527477936183934540244411

regulator

3990140741840927527195879309970.4477953840369748481843994091125341455689079284073641626492999

submitted by

apkorte

submitted at

2026-10-18 06:30:45 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-18 06:48:00 UTC

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

(1157421913805480030551, -36991618186498779582799124096476)
(1743883190967436075261, -2129467887225005475946739095846)
(1765885494604375230841, -3981162302871805974601093275586)
(-2809804219546275235049, 110667852494201910393978818420324)
(1535243623104610562641, -11832569367888829859924641818106)
(-2275400440270479429761, -134117456195342764710586591686556)
(6719276061723304468879/4, -3603464997606689480399082235133/8)
(1781576823688095194071, 5200845545488516494215181899684)
(2161871347491723354451, -34033020775586138885798483426776)
(809891274288767034631, -58148086993357741704518771061196)
(3153147309625604367751, -117283387013874062431875878338876)
(3804961291828602250159/4, -398194379121379373121300520271153/8)
(30097245136737127924741/16, 810263134247300167786589823892511/64)
(1200608847824260543001, 34230189538261884982993773305374)
(2033597621182715018851, -24155093668691131831179991513276)
(558412531201821560897719/529, -527896324031013546614236028403940772/12167)
(37403200351481687140279/49, -20866621357948038865648123969897588/343)
(-3079151741046592481321, 87102118964544729534083537640356)
(1730094834459397113641, 172059063337071660912384291464)
(1679808146203906059751, -453551970501963520419761848876)
(1677711435387347822551, -878543826166031566822854980476)
(-3198291702118311338761, -71446972416606096854315853082684)
(1863726087879187536991, 11365411668206257213313603909564)
(1034455790624310391951, -44695435257456863202207698202676)
(14865575375418639479359/9, 91326385280381013781239003903148/27)
(1755873931072363438351, -3173158458442715675987048682076)
(197693018337399474063271/121, -6340524472427208371469121099492556/1331)

Commentary

Specialization at T = -660/137 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -485042948797679528112T^8 + 12463578805199058894528T^7 - 230629938307243679610144T^6 - 1586012147509225586376096T^5 - 3868907398105357015696080T^4 - 6794745991541079206248224T^3 - 4791118686138149116505904T^2 + 6061871481875161826582772T - 1374235222299991775805627, A6 = 3820001086939854414872700973440T^12 - 155249980524531925200471635032320T^11 + 5279785372691676756614905714258560T^10 + 22439305421110286908829252745667200T^9 + 198311921848923368813090766393603840T^8 + 1955108933617356512048828761030033920T^7 + 6903605561322701279549833535328289920T^6 + 8139034537986456490540355512081445520T^5 - 1317692978359257302156310289643874060T^4 - 2476393922218195353910049397638432280T^3 + 5841882186136964852470754649206854356T^2 - 3899452237007593054578752344925650716T + 735364979113660346489805816953176854. 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 2026-10-18 06:30:45 UTC · history