curve #511

y² + xy + y = x³ − x² − 13475988212458140224877580403137155567061334357x + 595887183004541438412301782876602057287439402235546671638935245852589

a-invariants: [1, -1, 1, -13475988212458140224877580403137155567061334357, 595887183004541438412301782876602057287439402235546671638935245852589]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 371813998023695463104408943380209742606395805686060389724134210913039287870
discriminant (Δ): 3230050039767995941465304682565004975933777239625282300706909439707303788909135877913986698890718416408006853165474706617395712092723200000
Faltings height: 25.3888
naive height: 330.2653
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 47, 53, 79, 229, 277, 311, 367, 751, 761, 2557, 2699, 4391, 23357, 97003, 877867, 4503197, 4664591, 12394157
regulator: 47117307022781189875744592.8533325600483036021988483838738388934447825977
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:41:46 UTC
last updated: 2026-09-02 13:41:46 UTC

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

(24966012855176795075452275837474527/66609931921, 3774217406776553434710671397520215038141792461898784/17191290719558969)
(262684075838777977359938991150766457591/3225481537382329, 1137421125067983067658059808016602318735571655545985579296/183185814783091185389267)
(142254726856649533996084545305874989520604847/1844926778745632500801, 309237974540998836946914865123254599139019877219334302104881216624/79244421972360277165272033790049)
(7808913527165151922014264274380857/50112851881, 535631559266613710166464635736200264387561979153454/11218212909228779)
(26743071637434994971189413358004852443120837/348505994310719193001, 24035719553038071389780474701268218963289981797887715784099517036/6506019768655718343478980758251)
(52850426889316596683777191754549244654149332347516683/494720936056036764162818864689, 6742296106563866564322267576889172555665667545178775260010545683887317264088912/347968903347031295864275568134662354977204137)
(-38770025338005003106473974583870863837017/317329977833763049, 3657433076367654402002576593790121590983418576189250965718192/178758479556994472807863157)
(5635301423559174603280371592312576733/2509308649, 13377523591572050743929997332569493965918467855855023732/125698798154357)
(9288291515756207921916865340581874117327/354005818649822161, 3398692414338194619830179080933501934152370369339709072324016/210627755876853369311586841)
(99606203506739614386902185093224876371118624413/1711415560289784719161801, 6610807601243809484348227258757349622843360463348515009974492195652812/2238892360502551086705956962731042299)
(1557898410234055729378869149621599949537/18812253597057361, 17844950049596466146411527661703878975408478880202794890234/2580246733234612599787609)
(-78625995560879157562038003281375105713/718609297858321, 531239481510059658899457868888048750927907160709291000944/19263679682055948773369)
(825732080764770521154209597423191521674382145279/14065746590929919965808481, 140488965594723648072863762497557404125817724649126439548913949356820208/52752638127975512652491069100765354129)
(197238624811856892144290728557055789053/4485936210945424, 2824462640821124709457044448801952305624515133750884563293/300455141744071508043968)
(4765567644683118326860826313712103006750573/18641279563259464969, 9474838794275858070753388996340227815403768154481081179794541172/80484746298829440944710845797)
(-165299580979518469020951719304724473755673703713961/1933715233774852261197674401, 2849822260295145765060789243731037679636155982335684317668241225608137051528/85033249927866820445946101217037871728399)
(22044492163037832050645044692406903358839053687/412478145459248037330841, 1409927766063264294748388912692297202584120197138960952780174863473536/264911867714426887706733412588620611)

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=3611/15087 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 13:41:46 UTC · history