curve #302

y² + xy + y = x³ + x² − 1284727764113567728281797636015784768866707681415849262157224232063x + 560368321454261339256859338901915312332769858684945406858043869199456710681989058863306170127006181

a-invariants: [1, 1, 1, -1284727764113567728281797636015784768866707681415849262157224232063, 560368321454261339256859338901915312332769858684945406858043869199456710681989058863306170127006181]
c-invariants: [61666932677451250957526286528757668905601968707960764583546763139025, -484158229736481797117926468811255292357508238788175012972498868670847390044003856563630907590456883225]
rank (lower bound): ≥ 31
torsion subgroup: trivial
conductor (N): 9058040138868690262858791849730511364320517732812998655982537144987090191789878126630171056773171491466723174687496041392810438595512770829603147288102844437462050
discriminant (Δ): 56986667162894850943626331069759320443067423220527305082244725946223212417010574363800489149279120357391198820328667683955054948591887684940751932057558936585975074155693048247805850350226862080000
Faltings height: 36.7425
naive height: 468.2771
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 41, 73, 131, 167, 7547, 632881, 966509, 18145679437533309132469, 767028866604834801397681553, 30580600452196904409276223329355584892025407195996968868775951126238056443210297
regulator: 5520367374821893536678475926502746956624.072603230810343324240227195436655202216427344786535970988197
submitted by: Ava Howell
submitted at: 2026-08-23 20:02:58 UTC
last updated: 2026-08-23 20:16:24 UTC

Witness: 31 independent points log in to add more points →

(-343746913367434053557936715792581, -31005994945795167281668477280091616990476695982462)
(739387066521976456185309925500635, 3830813927243125109605210052773411116315836476962)
(794918160232596023344129483097739, 6435806665987309972375508567418781686192466975138)
(552229821725543064455253103966115, -4394327660511778936811414681478594663009127495638)
(621111741565922967463663441261795, 1421683191996488160880565213880200774834445145522)
(672709190481033101277453068263195, 739179814158401859498464392317452223287910224482)
(821399009280840681555588652225649, -7699942068198872209289633789793873375706107955662)
(765203485098820658379256355481245, -5034345308760298854369394902951465889284652680168)
(5399595603851507511243908479925915, 388654067273650918227195921742660550010310029109122)
(752668954339285868300409092384315, 4448429560131722871335630396999897920170009189762)
(665267573831105472345147950403835, -339632380726246052348178275348181915170728600638)
(2348702476455858806523697264250795, 102466166054736815527265002023727452468648322685682)
(-485301239340016399925702578908055, -32703997667874232639739344802658052943027850955518)
(1671562150318197588290891477490065, -55528472789639333688258829014227823359584879796238)
(533792265728561313885832952129951, -5165872676030535509137532850965718653070997396662)
(8050101189939018648004785213625795/9, -304090504548899861983493167851611965128494136836666/27)
(641126276756517817380587191947425, 475393761225519880788251703870437551199359754892)
(-28607312564278696457116932283279229/25, 2883492798944948226520935391601885431994599305099066/125)
(56549444812087071231733196839837255/4, 13405577698133557868624072930089297336164276021592851/8)
(1436747523773762064481823443661979, 40991896134294415817647888149334972652525806317538)
(-5064387018792502839844249474841045/9, 897570094862437631568619794896775730629375504169894/27)
(487104273264503086273608258117432335/484, -180314887995146178717423515157548125006198119570993099/10648)
(25599342024506626201702677512314955, 4091893764954118703319829997327215311497327967339042)
(980266989498097787981788009573917995/961, -525575897314714907955968902277850024344706090034709538/29791)
(5818507410725525932023407879789875/9, 1789182708478717961552943758166081786352797832214/27)
(19197683320190396803412613672365855/49, -3712724452088495818457900090458774587078999890353974/343)
(887913128901643530184527456121547975/841, -475948997371817062287059043149845391545859899478008302/24389)
(1816718417495680977851120075801270195/1849, 1248545591609034704022269662335921779441905481482388694/79507)
(4528903098191425769774358894938128668457595/3630906049, -6558125371552387528524400913253424783234119422079260125534495614/218787505794593)
(33880100347989327554317013369507240405/52441, -1625428840401701290033088199518465040555146308891387272/12008989)
(-11477003798992551861310315487329257523535/8922169, -244939776343182008186576362205394985355490153552116110249914/26650518803)

Commentary

BSD + GRH certified to rank 31, found by Claude, Levent Alpöge, and Ava Howell. [edit wgxli]: specialization at T = 164518/924945 of y^2 - (Lx+B)y = x(x+D)(x+E), where B = pq(L+p+q) - pE - qD, L = 446667T^2 + 471466T + 239031, p = 318552T^2 + 368554T - 72570, q = 733413T^2 - 45082T - 14960, D = 5(7174492962T^4 - 7114589515T^3 - 22069002960T^2 + 3909144679T - 205134150), E = 882769396002T^4 + 811447034567T^3 - 1174040743T^2 - 32493137198T - 2386325360. [edit cocofi82]: TYPE4, curve #302: G -> Q (one-point replacement). Verified by ICARM on 2026-09-25.

last edited by cocofi82 at 2026-09-25 19:53:02 UTC · history