curve #620

y² + xy = x³ − 415938132573567323342413822024408271176456864055389354800960x + 102591844757223519689336596996988088429820144015438414038723184271561200634877921842073600

a-invariants

[1, 0, 0, -415938132573567323342413822024408271176456864055389354800960, 102591844757223519689336596996988088429820144015438414038723184271561200634877921842073600]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

34363954940293776614841021425585547642251609013785254548912325067825448439331718049552978507578606223080831625486859605329767365039492740373364087010

naive height

423.4473

Faltings height

33.1332

discriminant (Δ)

58550180248045657853709688635053808813199645146411558409979534699726356967162824810800280047229294745218860453474092549834186108564691821612636577701641725452666815153465344000000

primes of bad reduction

2, 3, 5, 11, 13, 17, 43, 53, 83, 101, 63793, 6233959, 513130233266133554752239594636425423, 120861662647074154108853351065139542342270602097724977043568579755444358911911302345087601

regulator

392943196200900578899505896808255509.0281719344350607298833328021956077935961776540743267490338

submitted by

Mundoamundo

submitted at

2026-09-06 22:23:00 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-11 19:43:41 UTC

Witness: 28 independent points log in to add more points →

(1551960889982074464774345360420, 1787483743171596155016528390722405242320241740)
(-620105807367748223122275744000, -349381536992221025458175435479081841433877440)
(1820796764545405690096775794440, -2319858209903109927249328424865083836259318280)
(2555909528380166186603720437200, 3966914934438215260569714891530194799636277360)
(346452947248286778625388809320, -8571872768135084644345441476272285944166760)
(344005963340697050451271609920, 14708767503659609240077012280404409589915840)
(402936678451636314111341360000, 20373881711254747240358643662340959587701440)
(276423891724358759967029099520, 93478358015849088194858794842682351064706240)
(10889686431791800944107348444415/4, 35006446787981486561026876585698608811204872625/8)
(21407265722848868809077164315520, 99002667209385953258534549590814567399502402240)
(5343350856888414886917235960770, 12265407757827651045777329604092447814968503740)
(2686617982478048042882213990880, 4286829643633439795391082844284507911314137440)
(-2589687918032066463298928070792000/3481, -3448812613094585770111308851027414514841970957760/205379)
(573976048362682903860571540272, 230106213688783847242497262492100375656897296)
(347801902810852336437916201920, -180753846394808981416927475558225732396160)
(-653807964101026496899621848984, -308310853749411539548004481576129938413053288)
(59022247683271104828699581579520/121, -167117882615790947590308686089173202661364260160/1331)
(226395640520617600042998066720, 141524450934134822321973956614478333247311040)
(3091583061128823261409666414330/9, -424220042854018483128945085442268705408792770/27)
(3063688954770096945644875609599510/841, -167081114817484961756187577545514874081884730710650/24389)
(-2447742917585669472379005231736080/3481, 44706403556766251172348662394312708298532240315760/205379)
(9269974368767077060774099177406526720/37442161, -27862124932648595768142207479363937012072206528723019840/229108583159)
(818547695355219331599400625581470/1369, -13174961202133392434253190439480202488854473180210/50653)
(343410359517436992295394728320, -15905867508983039630864568645216026621501760)
(31605705791781456476635640377078080/516961, 103402667990027139512972941268808280548048048036467520/371694959)
(-5364607554964966535454409068890/9, -10056851915288516510045049935024393577854013820/27)
(649816207883349504542726921238231840/1274641, -218448655233879477543967917629613270243269529385494240/1439069689)
(-2631448654622699140522897972158840/27889, 1748859724716944140331638038043149652272661457583360/4657463)

Commentary

Fiber t = 45532/1617 of a rank-17 elliptic fibration of Elkies' K3 surface (arXiv:2608.25406). Found by a Mestre-Nagao sieve over t = a/b re-scored at 2^20, then an iterated 2-covering point search (fake 2-descent over 3000 cosets of the growing known lattice at level 0, minimal reduced quartic models, ratpoints). Basis LLL-reduced with respect to the canonical height. The 28 points are certified independent by the F2-rank of their images under the 2-descent map at the good primes.

last edited by Mundoamundo at 2026-09-06 22:23:00 UTC · history