curve #523

y² + xy + y = x³ + x² − 9101887567534335981075933733070899859721561907290x + 8979540953070920624854972089755397949631493850838066431403838505793366247

a-invariants: [1, 1, 1, -9101887567534335981075933733070899859721561907290, 8979540953070920624854972089755397949631493850838066431403838505793366247]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 672417291022846025967500274607123896588697929943810266449219368332393071485129230
discriminant (Δ): 13425470368869166359458471256690306126271090774376039819588530144271474445267271237417585372856717307414207292458646606967673590054032480665600000000
Faltings height: 27.1453
naive height: 349.8113
primes of bad reduction: 2, 3, 5, 13, 19, 31, 43, 47, 53, 67, 71, 79, 139, 229, 293, 419, 691, 701, 823, 983, 1249, 1481, 1889, 4789, 9349, 10177, 10487, 13751, 34871, 5931247171
regulator: 2287792295413576293946149.62792192428147249092881348688560385070613839214
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:43:03 UTC
last updated: 2026-09-02 13:43:03 UTC

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

(1965886856946901381902783861984097/704849401, 43039687001053231928577485308126690477249455147439/18713046747149)
(-3897998288554549445644874246063/1510441, 7256635366577209319559864099467217412987778119/1856331989)
(4195950884949480021762444243/1849, 8317842168408592342180922551313221307247/79507)
(24318861906034198032348145868337/10426441, 22304814216411489597498211105103163868882720919/33666977989)
(40905207843711009320221825038457186657/5753374201, 261618147697530908218471196024065141574808392854258723921/436399186520051)
(17755577175046369608497546697153/5139289, 50478192486235193158827779833515317835902779233/11650768163)
(13478774469301828578751913674468138737/11728028993641, 7720512929957893249743244905858312616391442737774870631/40164054380231835061)
(-140460626727529521790440112719493371743/42369437290761, 454817491114635798818826843413591611956720738872532615751/275790336193712976741)
(5473675075802638849030345347595345321/4178458881, 12806114173361561350634737110287196536131111591000390699/270099760526721)
(402529174011072617592215087732759383105/114094292709049, 5555636482973071403645213246459715856428960307581381657871/1218697388911657930157)
(7193522377393114937402002070905203/9569339329, 1498408381347757549117656898103879206962334729842657/936101481180767)
(452402469778931536357857978476601417/533059151881, 531661997965305971837034649050141467488097652060652759/389191284320685029)
(1206829797735185374718628961139873633576055613/56842038219319005409, 41519046849161559001113610716964825869911782807072257631780045265317/428552924807756545260487169327)
(490387191096644825533370915911989659476929/221634291687769, 343406311864177458786920123610561123833959847812021373231037537/3299555472826700269603)
(279041632441428032314540947286700424716233/87362702551543329, 91270314321180512229922285554909879362817298309295255472251077/25821953255366413216018767)
(-55147467718475641251212258817845276646007/23831420486758369, 15456144078315469698050692412652021154547318640641626326988773/3678958545840516155657903)
(34573334478388184114616590778652625917884697/30852937818472483921, 74154384472684086988700805600875508875579938370467950117697644499/171373941682005253627570250969)

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=-1640/4903 on branch u=58/37 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 13:43:03 UTC · history