curve #526

y² + xy = x³ + 107402872876637653862507156198746830835781010x + 4224475166675399254285361022703117514716762429657056963210155804292

a-invariants: [1, 0, 0, 107402872876637653862507156198746830835781010, 4224475166675399254285361022703117514716762429657056963210155804292]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 896646894085414738018884578533457083277325095657891101842582783442629320910
discriminant (Δ): -7788845956417443911554322278168134310294674003014547938309309575453945390838340363818467939827696940904643050116642469355103174656000000
Faltings height: 24.7548
naive height: 320.3462
primes of bad reduction: 2, 3, 5, 11, 13, 17, 53, 61, 107, 139, 167, 229, 281, 463, 491, 587, 877, 1399, 1423, 2099, 2591, 3851, 4129, 6421, 7369, 7417, 34303, 587413
regulator: 2636742048426089067919863.66574613493253502124361544397742051826238595793
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:43:21 UTC
last updated: 2026-09-02 13:43:21 UTC

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

(-3761780732802174685862877684/299209, 154133708956120498730798871827429755609974/163667323)
(2495150654599314112273816224/1207801, 2801723577767260970148696186264239812371162/1327373299)
(7236342444903426893422533244956/49182169, 19527188493178507103674125021074069982942638586/344914551197)
(79011304842620331736185055656/508369, 22271031337869520773394676411145762644527986/362467097)
(3412512168014431686263886661952598492/2799287033449, 6304164137163291246087488459597691249359250993491550678/4683506730672756043)
(14733067881848492130956412628806556/254866454649, 1835955309628010954314988065178392877369751164528806/128667545564365107)
(445426823641567222882703973048/582169, 297307889363175791523068213213648523755438226/444194947)
(95773541100153994221846293875386/277455649, 937748214213890889523914919657725562062386014854/4621578745393)
(103412599381077506056053514400834685196316/5196879063984274969, 44708001123296815860915620805349465670200548981700005307820474/11847150779312927142608515453)
(-3036564086541812289781941749831724700164/293152354096798849, 224500353473404608576053379991947732946769758444734888981954/158723078811026107045388993)
(398508456053080731781071996924984/695324161, 7956685151686935029433521885834051173511689934022/18335002801409)
(33620910308209498670379354498204/222337921, 195522014634016413474177807642905638677075075678/3315280740031)
(-72519749011202448198365672644614369077891652/58105995124594852231729, 28321266369832639248338415796526239881556215685820929742067592912110/14006557709602424186191243506500183)
(-50097934008999194587954403602316102439/3704865703641664, 123436519520236967908284941932665813216650377370351985081/225506313341904976819712)
(25087654751338468512443484/841, 142156620745291157391934390906869054282/24389)
(-836317223593929460453632306836729871684/67474505856653809, 17431688731966755260197124973648973669230311861542669305426/17527080005714418847778377)
(-753714659100004438091452920433870634244/199288260207276769, 172606670313003313662438767540650598103927204206527810537314/88965694559443521783590353)

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=-622/2197 on branch u=82/37 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 13:43:21 UTC · history