curve #510

y² + xy + y = x³ − 89537110216200447100759138717171130974165931728212063x + 10309659961796079070131625494220076685011127717486239262727191317539981936305838

a-invariants: [1, 0, 1, -89537110216200447100759138717171130974165931728212063, 10309659961796079070131625494220076685011127717486239262727191317539981936305838]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 7007913908914431737986612553664370128128260957329061089431076964775290140500425316602770
discriminant (Δ): 22923749481637810847522061234251954659371561228027020443706000449553505366966302593000215673546427984304191609969320877843951964763047225162260896690519312500
Faltings height: 29.1745
naive height: 377.3931
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 67, 83, 239, 307, 313, 941, 1013, 1153, 1231, 1237, 1451, 1637, 1931, 5231, 7321, 11057, 11969, 14387, 61001, 93811, 112663, 513173, 580417
regulator: 34124710194134030283263937.7129188174111265903532983351012941909260500657
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:41:41 UTC
last updated: 2026-09-02 13:41:41 UTC

Witness: 17 independent points log in to add more points →

(-15626394001093430720231568238371439/45265984, 85803708088939212569670198948135808031068074402045/304549540352)
(3455156863565747838760904768058215985524/19020859186681, 16404935786248860179653841335624471801237465102591926787410/82955501983139165171)
(223618878491111911510228841023893506/1257908089, 4586483632198685668626381076048487305025525251631880/44614226192563)
(660651222527920601836495301900903390343581/3886780819778704, 9149454148003734166746454214487086749201558713227073169162635/242317665540886150668608)
(33647931792399872181246486952280881633880381/186510493712120449, 13524928106511732099235286755879035663169741194620244411294540600/80548027110335717325844543)
(90647004013196604298691470721709261647492/536425945478641, 857352594618060393155544460115681962158364269181971653398362/12424092124283837456311)
(2142830498807766757730903782319608567172726/12193408318470289, 60942306079528447347901939663573814956876983751111913516678220/1346442079094624988958663)
(11151557313809957451242549051909482030130171741/96572036254413485089, 1166261501735519074144004141982544013654326494037446342954029833340250/949023754436132407420233787313)
(38576637752828280145679439298970759/3147076, 7574576223745290839631284025329401244595902082963735/5582912824)
(684113497483507685306801320394645024103115821953/3804621170960565796369, 36015267612621041037680385902484574489658129337900614482762198872653976/234675164161480391857220254016903)
(2115365980919352540102496950483530213432888186/10464036190672960009, 23223374797847656208335308288294274108132816942674913453502881115145/33849234341569098044732226277)
(159154274490035701032786044586303507844637/649424725018969, 29070735940169998191722746707675249276878377293365564628208056/16549818249959827448797)
(6411433269938222347419229544938763720529410813/71103009770064763969, 1033116817465160999137765666557864201655699125374568832476151402114104/599559069643215922139320070047)
(542330106871978367862959904992419876210684837327331/1761994452186272607814564, 8071679069499505283340547275381374060108064477081413542047736606466468763645/2338873886339938593289171752567399512)
(-109594723283065934287890397180541738469373959595762327/1142583433484351565551782144, 163928407841925265080798729913297034345706415983530738891057683849524500154938541/38621744305904412386841402651490901577728)
(284154264372503388465413314494787277031230261/2015547309165075625, 2000393603008215230396006043179445193980307964606907326947476774066/2861471960546147844416703125)
(95867407361888906344024224369984644367000758217297374/578068332396972376634812201, 2056827915468524192195387359874713565977549744263532338852730989041878101434055/13898526770073301311542032169400533627051)

Commentary

Exact rank 17, unconditional: rational 2-torsion point, 17 independent points (Neron-Tate height matrix of rank 17; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 17 (PARI/GP 2.15.4 ellrank, effort 2). Fiber t=24693/1657 on branch u=29/13 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).

last edited by Matthias Breddin at 2026-09-02 13:41:42 UTC · history