curve #4170

y² + y = x³ + x² − 51616082301891374930060404147161154643065510x + 138386033965838080496617990011875803172522629686647313777103566814

a-invariants: [0, 1, 1, -51616082301891374930060404147161154643065510, 138386033965838080496617990011875803172522629686647313777103566814]
c-invariants: [2477571950490785996642899399063735422867144496, -119565533346484101549092808801963638657039409445657661515954684594456]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 1598740076010203977106890461848627662726267706800893427358467211796174691199865947728895749564939904268799490595
discriminant (Δ): 527962185248660500083241120869783059790627612102857686242563329967356891354889462319991412244019387187824473441903646389181203703125
Faltings height: 24.0466
naive height: 313.5708
primes of bad reduction: 3, 5, 11, 13, 17, 19, 29, 71, 463, 11497, 520609, 1480708481663784570649, 169564185095878422973915902109, 1610685729850385606820172851348616368497
regulator: 4924808336656359263427708349848.0312849730893512297283929372591533195834336529714283394476791
submitted by: apkorte
submitted at: 2026-10-11 22:13:07 UTC
last updated: 2026-10-11 22:13:07 UTC

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

(-7803737102772336749029, 256807218885549347883089560527832)
(148598048125769383421, -361551194347124270399241845388518)
(59557898731382083603136, -14433455655375802200892442917845563)
(-1340577407764471820092, 452959349231760737557512521557714)
(5260930434424729003076, 111562690906038098419712078505517)
(-8124159060969581018089, -146672668468465648114635877414013)
(1241737160137366936466, 276056289394632472059323573309797)
(13268203510625128496069/4, 484630877859994486919361937310771/8)
(-4376061773864080189324, 529584724501018490286317181617842)
(-7958877422186966394697, -212243079763449975619643687155805)
(4779566635663717714811, 29481838137732978345649078222087)
(-228389056879248735001111/49, 180754400564413012639282396180387666/343)
(2205538037585240816582246/361, -1550227233053567024871858058486058042/6859)
(-8255810916337143268159, 42610706278039566371549587017577)
(3050292824153319571601, -96553955300294650096959013629818)
(89670972262199219237921/16, -10147716919238101542410021635517447/64)
(302572749068939381476577/64, 2813132508844949118827671040634383/512)
(5378089509268824768986, 127848191650653063972983415248362)
(-7421342415673724068559, 335717740139179277947536954630502)
(428199877112315019996419/49, 204271198275261060004689333999586336/343)
(2317077833342976637357793516/654481, 2503678962297681605906896998993308017298/529475129)
(2248323890709822981552401/16, 3366904537097944528540999561330881793/64)
(18739996363683750579352844/5329, -7388984982707763626948425062822012971/389017)
(23591440283962171999324/9, -3921716728790249625349836207715601/27)
(4792698034793369160311, 33072546298063520978577686783587)
(-8147295863158843568734, 134583879410452676910137436049477)
(2813394955351542688301726/361, -3139156257981068737178797173088593002/6859)

Commentary

Specialization at T = -1163/397 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -8601190111439927121627T^8 + 70553802194275661608968T^7 - 40117514965744884873804T^6 - 394125757357369375667736T^5 - 727539205447352323463130T^4 - 3371639403550126915091304T^3 - 358500560703347306973924T^2 + 5575527756032095197563112T - 2574982297007819942866587, A6 = 311073935845060124132541860022390T^12 - 4299424530738788428378210804267800T^11 + 17883941445167312163435648485037300T^10 + 19652059401694206221665695004579800T^9 - 317465301013995782557083261357635310T^8 + 112305603376549774936723074091082640T^7 + 2598376408341179206723494257772920880T^6 + 911887687830002593185098844364676160T^5 - 4509956533653094383564183143089914510T^4 + 1581320920924061257174062701329500840T^3 + 3668404075692271898587012925610059196T^2 - 4676936909330964399995425467736251576T + 1568898839279682527117596188509969334. 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 2026-10-11 22:13:07 UTC · history