curve #4273

y² + xy = x³ − 187807034265603588688491473651872808360x + 990423560671547949782589371323389658970092939000208993856

a-invariants: [1, 0, 0, -187807034265603588688491473651872808360, 990423560671547949782589371323389658970092939000208993856]
c-invariants: [9014737644748972257047590735289894801281, -855725956420217428625679323290532123735731685399115412893505]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 17134222746387774970099915813476227048841598658950608099457561222375948826440847228073750929265414
discriminant (Δ): 185302508309072474721711085693066416029736402378764218412967062072319829652551831902045227662998214205891613622272
Faltings height: 20.7207
naive height: 275.9990
primes of bad reduction: 2, 3, 7, 41, 73, 1471, 36187140582315791, 2560598347654085510138906761380200366284296076666135789818938328782176279
regulator: 1083616344002359656365285397.461952016202985813751003441369110760550297372760869116403215
submitted by: apkorte
submitted at: 2026-10-16 04:14:48 UTC
last updated: 2026-10-16 04:14:48 UTC

Witness: 25 independent points log in to add more points →

(10598949031235323936, 13803193514854316911692683608)
(8015015501234185936, 188840089745516162486603608)
(33256609107063808720, 177556966221608483869269409624)
(7587855471947503720, 1498627768180368134460874624)
(8481568266835779712, 2768385303986588671104136024)
(7448624967717269212, -2187260892342016395883901336)
(-13592558631541176320, 32122772194566194779582188376)
(5245768336684499878, 12230489284936059737200174468)
(7490374866125553208, 1982754082348556875628103448)
(12789164471914848208, -26083694892711636363800271656)
(8403087911186816688, -2371166543017811292519121416)
(-3152574069415117520, -39384852235147303507153274024)
(511133146790119041568, -11551716603063282838234346014376)
(6590391547267943662, -6240501908010722097445248956)
(3979756358617224712, 17493728486752802484637303264)
(8338018774144176112, 2040923608149736557711667384)
(8963014444451716756, -5211122084512894790885389592)
(6480367841454429616, 6746037906351559726340208376)
(7786136362890277408, -398102691451671107367103352)
(9072873840318107836, 5772880451922783569776494556)
(-4177140832302731312, -41255728414362650969759925416)
(2978481322104012076, 21388477507807540544746177516)
(188676800077588249936/25, -212819984096953989635313737224/125)
(6470777701925388112, -6789969938998026271499318696)
(12084447491830344766, -22036741254008064806236610492)

Commentary

Specialization at T = -715/1029 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -9801785533668367408277499T^8 - 121843241966486333706286824T^7 - 383434732762042957835672232T^6 - 85754468638713501529031952T^5 - 584201627367901316547607440T^4 + 1235140044591109335942134016T^3 + 994771523582906627448194816T^2 - 550790635100880788122728032T - 647714191600162651688347312, A6 = 12142329458953725303990225302566830294T^12 + 247791101750863056653001398463488667192T^11 + 1886090538602583189200926091842195641480T^10 + 5912076790668428510577901681319655809040T^9 + 1282952245208814516295621371483268940160T^8 - 25791045529140834460411754210526102555072T^7 + 619130399100809058015628676691838434624T^6 + 31601519751505535243254509940106610677280T^5 + 10405299225003196422075679041028274690640T^4 - 27172444692997266799042742904489658895040T^3 - 13432438015701644454859263568215615214016T^2 + 8212424024664720375926323815936672405632T + 6370397708074310285430666560056789129600. 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 2026-10-16 04:14:48 UTC · history