curve #4220

y² + xy + y = x³ − x² − 50477817923673577716400396642954683830x + 143677006664748416468007560110018291819554894594727481797

a-invariants: [1, -1, 1, -50477817923673577716400396642954683830, 143677006664748416468007560110018291819554894594727481797]
c-invariants: [2422935260336331730387219038861824823825, -124136933758342631817455323263542311345352943254966332565625]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 91845577056807951694322697837878934816027414415778863367584405899332338321985650038450
discriminant (Δ): -686260168613455500972898141221570848955361986552665769620352183612024142080232816781348483284964812942239744000000
Faltings height: 20.6053
naive height: 272.1375
primes of bad reduction: 2, 3, 5, 17, 19, 23, 29, 47, 59, 73, 97, 241, 367, 2832287, 44870469257, 416713987128255161, 10300394492972934361302585228509
regulator: 813980174066564711182061.29554247208719573943227063582365410320272198006676060702662166
submitted by: apkorte
submitted at: 2026-10-14 09:11:13 UTC
last updated: 2026-10-14 09:11:13 UTC

Witness: 24 independent points log in to add more points →

(1506024575133016489, -8430421212246179046705327445)
(7165888759087265869, 12244455558114743467442370515)
(3424137956171178139, 3313767595577215130785184705)
(-4966967544826333511, 16488171309818521530880172555)
(5618450972817430633, 6117798180765633803165355371)
(2647204251992306569, -5348149225283893367145796885)
(-11292616986686933053919/1849, 1190366168174551994313962247264305/79507)
(30946507026060565033, 167983959284307502442634675371)
(-8177968677974126951, -3089862326588594417536364725)
(45096680592210203239, 299300672111036551974020074205)
(-6161516419139636951, 14858645560558164503850735275)
(1267353073661678569, 9040983959097836041490990315)
(3739943337606114499, -2684040640074211074298204375)
(33108290181854335969, -186452109167128090072768508485)
(1622449485435444674449, 65351057445047606029200605097875)
(-5469717006355874111, -16004201438293117033389158845)
(4481479200086034889, 2732435065001324186021790155)
(20171373818355448469, -85632270948267947046288184985)
(5564667304223717819, -5924320441291830523978715935)
(14975444154858155071/4, 21422802247893133780078601575/8)
(2375524262587709572849/121, 109106687925795750252657938048065/1331)
(1470469077997949993, 8522353027951557742525840171)
(583586287446770650849/121, 4689278495295274761343411029265/1331)
(363770800274055073609/81, 2004477520805778771721681994195/729)

Commentary

Specialization at T = 153/172 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -46899334113383125285632T^8 + 270055671557662617610752T^7 - 110499855853494623134464T^6 - 1097978606080292495697408T^5 + 798354426662958846861720T^4 - 652775607268414350434100T^3 - 1472431025874707052625953T^2 + 2703862379229266288830500T - 984764302175595161629827, A6 = 3885040300642071987577481657671680T^12 - 33785823226478031065590325732302848T^11 + 71535134823792731310067418119643136T^10 + 161716991851894171424472252351774720T^9 - 720321276317174942738930961334015104T^8 - 173535940015248246031810704538019520T^7 + 2662884349300283514864711968426262768T^6 + 18543461312481910222042123064099856T^5 - 4146251221815493809167896322067912459T^4 + 1352969495572413131622726277427086950T^3 + 2682293599935058314574939466346897591T^2 - 2185595594424745389481946472892633500T + 480683366018569365299225874952658046. It is a 2-neighbour of the fibration given in curve#4205 (itself a 2-neighbour of the fibration 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 2026-10-14 09:11:13 UTC · history