curve #528

y² + xy + y = x³ − x² − 374024585010802945575452832206743652356394671952x + 79533615237633749172768991731343692332956194038170291602659759576822579

a-invariants: [1, -1, 1, -374024585010802945575452832206743652356394671952, 79533615237633749172768991731343692332956194038170291602659759576822579]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 799959445620234844233764836050816395509063143013902825458669780365231524648255194970
discriminant (Δ): 616074787578024409641514665956186314751259819860265274832487459076749882186490124563532971697912341040143120970541122256182623175593574400000000
Faltings height: 26.3251
naive height: 340.2356
primes of bad reduction: 2, 3, 5, 11, 19, 41, 43, 71, 103, 223, 311, 359, 421, 647, 821, 823, 1297, 1733, 2063, 55381, 122839, 169243, 292483, 655261, 1073507, 214659463
regulator: 395351602368354651682517.3118480999025217616033483481053452939660063292
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:43:39 UTC
last updated: 2026-09-02 13:43:39 UTC

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

(354633988358131449619271103/1369, 264555143232733788892223630728062411643/50653)
(-551689953288724258781091, 343461600955400443753544915877486673)
(-347580083058768283628663, 409322973226376697539940027189395581)
(-964684991070760344943036083/5041, 135865613573767466872592784898134944709191/357911)
(-4136817643797561959143236914915421385624882563/28062063475383244912081, 1704466915452137099649935896611656739712969848158704847137084715703799/4700882586351224550655164817781479)
(928147996606156310442543350353/3594816, 90310967530710161449720544718587470733882603/6815771136)
(15357392991007407287383761408564192995877/2171556644801296, 1896249262446992531877223732441560511629011421540968146074739/101194461471701180753344)
(-6606583589703211818592597304878705060163/17278379269668001, 927150979937301251341921630692853702544876555227401530471169/2271195422176487874779249)
(1798279841227521606650281597885693/1237843489, 70253468815074350414403343163989272573781425480447/43551047473487)
(796128099908586356092810785786609023853/9101003803369, 22462797459364817091966521952980228160941106256075667857643/27455789960954155403)
(-175039495971445347277824860547526252209584907/251263072344072507889, 178625014494978145408325669331399421974141828764386465141038326497/3982841271425216581390639102537)
(30655127685485349997430334701713441890028545933/133405510852616526124969, 3685312005286623109451637495102743802980230307747596660979939465388493/48725988161136409409916464128690453)
(1694407830921012978610074057634085365299813/1743081865176678049, 1833124509253927231864617883391051450301335229882053736696895433/2301318225126469958022900593)
(722489351463251009710355911883948746436809/1410576668205770025, 250350609677808075823119313986164823749794573165996579486425113/1675309472617507667010775125)
(7275896368033934553232595249768271745680013/9136676641798670449, 14787214571985436347392984073277072205628806728248748473413250983/27617374074391070147292119543)
(13376404297078074209933352188456814702091186745709/119338303881000319999097041, 257515369194219109514743759974126712671870906971683127783612144618886807703/1303676363585851110273652546032357812839)

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 2). Fiber t=-157/648 on branch u=59/37 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 13:43:39 UTC · history