curve #519

y² + xy = x³ − 239498035995170389121184973401135586620595x − 28313506560239172481237594672791743924212306882379634187314463

a-invariants: [1, 0, 0, -239498035995170389121184973401135586620595, -28313506560239172481237594672791743924212306882379634187314463]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 144709965057296981296158253023483904833485520354172293743617612639487430
discriminant (Δ): 532881471847975695029613638600591195746203359824092309322330797911276879056397307228060361517620266723065553306624000000000000
Faltings height: 22.8261
naive height: 297.4517
primes of bad reduction: 2, 3, 5, 7, 11, 17, 67, 73, 103, 167, 211, 227, 283, 401, 641, 1249, 2111, 2927, 21149, 141689, 206279, 1251697, 4221313, 29338571
regulator: 617805698060403700269060.741775445861235363416455499189386143963329899255
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:40 UTC
last updated: 2026-09-02 13:42:40 UTC

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

(67082540946176980152686/49, 16123588269568950299285884749369599/343)
(-330878192001158171236, 3834905613688862127829596701243)
(3991250292190875338566574/5041, 5971852659665526532196855483235650623/357911)
(-1640426409731912955556537707226/4485114841, 966600753450013937647570085375689291498078723/300372626016611)
(893211888284266062238293477625006/1526171156689, 10663651745561916434157924725593454291067390937231/1885405902063926887)
(-962606148419364293739577033906/2444214721, 268145583136572942577563968854265352964839367/120839531591519)
(-193497301751482070527587252214/1418049649, 72159800990354035431455585155661651889586649/53399495632393)
(-67822080430218074967498771137698/189668089081, 281385649413175851188629154227791904891587775045/82602159807577229)
(38021076259578053289210127736/52461049, 5080777738470239610609889823269694465772851/379975377907)
(-50752235791028341431066263074/183521209, 10180863869476930552618532289245278392531739/2486161818323)
(-23097704949766730580513346/175561, 70509206501662163917844587475352342787/73560059)
(68723916630070515882706501078327136/76475777071849, 14685196790390158717652826808293780032797201974103451/668783958951733594507)
(-156480279995244315883170873766/429276961, 28892695163989254330952356393162720415114847/8894189354959)
(-1729815191418444042571894/12769, 1850687764240362418271751164096822881/1442897)
(-71686243249330097920266282235721074/302981124760609, 20498168369937796428829012357350550023392825275778761/5273796409920200748977)
(23479983006374107156150430703029006/43313865080929, 306488699331159523067507542615824841949550101986519/285062709731475212783)
(-509678515252760949437014560900706/2257960027801, 12806269124173282759768899848056741517609919706123/3392925893735200451)

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=-2346/7631 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 13:42:40 UTC · history