curve #527

y² + xy + y = x³ − x² − 7542656338846519614190307528035555377636651798585237080x + 7846844307703241714083711878936476316769456277937452916435814466987255176116083547

a-invariants: [1, -1, 1, -7542656338846519614190307528035555377636651798585237080, 7846844307703241714083711878936476316769456277937452916435814466987255176116083547]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 43474516028660989841423831580088508738112258305978774012023297879965482051705634517491450
discriminant (Δ): 863792463438850810499813470881430826697895035564996350280399616447126620659706126257255255657128934440745210076146169227071393499352344116440534001610452317696000000
Faltings height: 30.4434
naive height: 390.6941
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 29, 37, 83, 127, 139, 199, 239, 461, 499, 523, 2687, 11287, 101063, 139753, 166667, 739111, 866311, 3589133, 15017903, 1560594379
regulator: 3541884225448180621905489.696566577508788510470906772182694585642486608
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:43:32 UTC
last updated: 2026-09-02 17:33:28 UTC

Witness: 16 independent points log in to add more points →

(92940306327734770762128439177457302244285/18110190938449, 24889731165997848983516480048536212599944035891578414535440223/77069855329000133543)
(2090654673328086302410257223811570461/1075380849, 812598580625623227177906068206999726428444502192757585/35264964181257)
(101678767598580702295165163543215474829/46877013121, 417406457100652148345488700658063296871909730411821897975/10149388987840831)
(28521341687361713996721521075474502701/16199434729, 10239573424128010612142779015372294410652232812776741645/2061815454002933)
(7077414621310719774794854690780053501/3877926529, 3054755764441133895789608200540327298234148585947834905/241490118740417)
(1346913455938096450860181479850326782905377513589709/1042779407036766661034161, 17146772383727510280972531516608146699730325569629073576811397070569780651665/1064850573435988074081903119156244841)
(175692747284836873455820352773636415836117/86636760404769, 24066807462404931002037789653358699881855523498522220762924255/806405175893664113103)
(8128852910775210377772898773951013315189759831709/43811547431310114481, 23173764135804277927279282750426233960808382244889531747389756461328911055/289989911350722715123290814871)
(-1074294045215520663283959833759301889499/339489179649, 1034971291783433955627076362635198426272919943287383227425/197805746946747393)
(1963527626836207925931829821724508498837/1036684403329, 19899584269363448412988933987658725954332986949119795799825/1055528215728311233)
(1277036701014875623300545788287972711214829061/432345284026080489, 30270116920981433884467051548163086709100701598802788898541741902275/284279857552213667664654987)
(39646052283259200587873062983655361393228878181/28404361291874569369, 935980733110694194553886015494885895944261201226432355309793792055525/151383151477910071339502081597)
(1726362226669440056354070553300877595071069765/919860723461502601, 15318361241028769519704789885073492601194558672666544356510995766227/882232623829508613691211099)
(29556395063897067832755102804893896278908021/10312117146474025, 103527418792160928863026755535499098567175547399004707178148134659/1047181007699846442744875)
(843693060945041467977571769727572321/112296409, 728984728480064987949607950336293071695702612503861745/1190005046173)
(-183596488959116181752821131624452561875169585091/238283673235930846225, 422615130314897184402013524105136171093683929230553907176590348429346753/3678251573655062606176307061625)

Commentary

Exact rank 16, unconditional: rational 2-torsion point, 16 independent points (Neron-Tate height matrix of rank 16; 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 16 (PARI/GP 2.15.4 ellrank, effort 3). Fiber t=-24179/8214 on branch u=38/17 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 13:43:32 UTC · history