curve #524

y² + xy + y = x³ − 20472874387656550683739132464643309550179386271005605498x + 35383097091233781491349920937186763913012343517808005359500828902307049381366178256

a-invariants: [1, 0, 1, -20472874387656550683739132464643309550179386271005605498, 35383097091233781491349920937186763913012343517808005359500828902307049381366178256]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 6078327483996412818131815249763988409748741635402527249657164415937279710460816261280068036185497970
discriminant (Δ): 8333925598473835484690119703479132753478801577489354337282970030768505435618465956788783134196355892763234781586547399720969122053185322404328461313086598542519687500
Faltings height: 30.6609
naive height: 393.6897
primes of bad reduction: 2, 3, 5, 17, 23, 29, 41, 43, 73, 79, 83, 113, 127, 163, 277, 307, 443, 937, 1307, 2423, 4219, 70379, 89237, 97423, 917173, 1012079, 26251997, 1287242582947864158632059
regulator: 1258140407261367080781112315.27737744218830435933431756192011303353255126
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:43:07 UTC
last updated: 2026-09-02 17:33:26 UTC

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

(38610875561436907335054758168924431268445/36819469456, 7586831505435701095762626939079586103846118601092334101082383/7065067077095104)
(27221042194289563063705916770524248729371720616385/2137702903949757251584, 134049751845543430724984214278400218751903738543927901629951154822480426079/98837333638451859999924842135552)
(507436402279777605298774794341832734895/229277211241, 3317866865610265218331503434603846144490969252344174316503/109784577781316789)
(1283465323765969556230877420971095185539744597276033389/558411743964033829208805904, 286005779237045921286520271908663572373272293825997289242931456805562322463710657/13195681210572358553282701110334744710208)
(-23491628017446253538490069990360038526920280/7111565934670201, 155194315925773957831686928661116604985369923251431584960913791552/599719073538897452020301)
(-3046813024619972756970178842908677431012270/754073403397609, 4728345153884264822347474672855450785587308291259117305937192021/20707155024439491990773)
(1347573447376993482574225884908335066794715483470/5886472890462113401, 1564027784057571417385183745276289675791676624675678634435778373304350652/14281792614002905442486348851)
(3184262731477940702190021696745660829280/751874549449, 102337177429552355980834985814827834705675781836884146293936/651955684949074043)
(6196556442061617873956063747106230605195531905/2175187926353756224, 42999888177496139838039681616607591506183273548228860699055757676461/3208079898127118447047802368)
(-1114135071612198781419047583296496400867356189670/248518485910267995409, 754233475862716403276354806807982810564918926814961122867933749432413996/3917761991165649728613318375673)
(1450412949620689250583323484285329301120954325153719195/246671484433717905819366916, 1332482120878567357003580226180234643452136016220591852679905392831845066689467603/3874167745019766827234109147524583891464)
(1372067626530778342642360463013651534785191701885622212286611/8219759337925845629205545113704721, 133248511691555816922889354740137540331331282811564100136426047212779424153563068003682486549/745227123751125844368586896464778105231404131743481)
(852098258112165186827085971418300005373025803/303994587762989764, 754808320276103207270170265830197669549366156036694173037176092339/167609556693011433355318088)
(6806521313939337909289271173646985279817041765008145/2433951936567441077578816, 2563319358345172450065744645611596271829781084287886185702551253958015574653/3797239561340976143509753412197732864)
(1828495210539808491281898056026736873454348609447953217870/653514846756660066008448563521, 1158275233127371701739950702288992471510341628062701924400938738670122278024907474188/528303134833650363266862481907294629239436319)
(-58268768509669919423702575941265611184418569134/28120765920178525801, 39144960690627496115241803029931800410439798001942686654691798410453739/149121656025499238684868997451)
(872687226567202586798453942474371491002762910295868115/395655797758080006325894681, 243490160419340018823872963398575656963134855440506822288651130035138509860223847/7870028427612673514742995197572653207171)

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 3). Fiber t=11511/4292 on branch u=62/29 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 13:43:07 UTC · history