curve #3762

y² + xy + y = x³ − x² − 19437117124017871824396126801635204001227612x + 32811883327559579598478031200165053487840164858566722434425599135

a-invariants

[1, -1, 1, -19437117124017871824396126801635204001227612, 32811883327559579598478031200165053487840164858566722434425599135]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

3174268000626664447890199710966144968041105363299985047543903707586610176360119017465386409590737398259154 ★ record for rank ≥ 28

discriminant (Δ)

4876130810587559530809237100161753709458594464826856621387065289527239402714075379609078894389508552780965877331034581333695639552

★ record for rank ≥ 28

Faltings height

23.7242 ★ record for rank ≥ 28

naive height

310.6409 ★ record for rank ≥ 28

primes of bad reduction

2, 3, 7, 11, 19, 29, 31, 37, 59, 61, 264269, 31343423, 1992787623241301574051210793060335615034084525434701774079172167997164741244609

regulator

414712471196139643006893396360.0060472924234499548491848395514918211640525720938933070341246909

submitted by

Ritik Jain

submitted at

2026-10-01 19:42:00 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-01 20:02:05 UTC

Witness: 28 independent points log in to add more points →

(-17750395349344523060833/4, -1423900166717069213900657335158889/8)
(2995893456954080622401605/961, 1493954349496369391889909009388892075/29791)
(4083055657741700772823, 146693816163456384642462165431717)
(6012914438374262316885007/2209, -881837700050419790679101116976187659/103823)
(27340242870829431132853, 4465190243340253720220786493880037)
(1875906925751383848145, -54323359920007408885045449375091)
(1646427200981291561173, -72616060831366097055243628953883)
(20789565851368042701971, 2934978493806489127554218582102221)
(2186301623141453330623, 27691480063937562458311779814667)
(2593684991417824238379203/961, 103656052603639970139335113974855347/29791)
(-34204796983901709808571/49, 73597241321285289606469713587253203/343)
(1748643535490112880981, -64577211615188397584911296260923)
(5481959428353082340965, 301664678452961755021189910649605)
(185955872790259698355045, 80166673732930832333554890356354309)
(2764300009111439418037, 14312816287352837741759700328037)
(4429634088685232111333, -183383021543802975913828036792043)
(448997988298303234668663268753372835935091307102872942807130442715986477/187812125435129226643543304136867013282344265007521, 7067467162630810421424234083603495107154302722510136394493318674627431947430546528833272179336384350750925/2573863085894593605586438873253111369718389674624693097117914791087962691569)
(7022991167975866252123985496360215454272932304048494882785416422017963742417631581/4117212650296746488618061213077215881198695454186238965663025, 567833625390201179667479805732561609255257297360465117451525948581733776368869350193408767489748450956990642030371593879419/8354201532205845992280118696242727098978129865415381475163208366107868307315050336905852375)
(38996471035102341593864511355868705344090499407499/2582829038213990673603667609, 234119131188951852165584854488328438688270588836090068049016376076272612077/131263352080097978912184664100056559724227)
(24733100762357889992103893898306376991869697512485773/2188307592817022540454503789041, 3629271512934370668884403892996516749699898927345949239409868661810908442963157/3237147965563533076560144597924947876911385161)
(9104922745665790298585261842433588938672143996949/2801611008491848171992046249, 9341101229192903046910946025576289158524129400569877994272918400261049569/148289961645539675525430365313118925169557)
(-25731600939426697170891344435779578465593231939861160733188377/15767327105557171138354699701616394308249, 485718974925139356516738377277112186500462406872413195270866341129808423297341229840845229919/1979872011360748278761128371614306036741669193139055075018557)
(386970182439164492275217235670189714597265032167696877/471061030015969676898464475176769, 1348580980329904516152897779029784400825050776775821751442358930390474319099721973/10223880532483451780343356800164018224301763740897)
(79652164065591363052372685287972254455393704929835361657769634957178992057/48394622023878799067428817365213790271694425238089001, 773533138160710563770479178303909805473282964911510094875578789564858103288010958232664855800178596359157585227/10646225317180849385128787678187044481813130074793375574279658358167862761102251)
(3136500125995999844579810421909048488701108128290557363173709211166012508453/1556490053053286907544998093165320299710346207208615824, 82993885800915183457885205898048436606514847431141293459546767526818831987190562048519375150229396530520264962615/1941867192270076073285777814066663445157072533603535128013929825323424336935936832)
(680557613878598183324961717735367514185775053361642867714852963309013144021835086205/286770845701544451737306259067189168295338848424531721785015609, 34279425513513314130525680382335432496543910069016012094385160487157736413602042127868304889510455024566835591553430860603615/4856266319849592829089519335087521832855052628226667685761211956101474232482989988647974990227)
(423973301392146577038465622798511483637110190608118427941006550419466771129534894733858476720069/180501264282834588335434850234939174139936131072266431279122258487720572809, 26095591452910507832317599694561333274650264313707537773901285188430152969339726303825165759977112696828495018898939952790052722408581028337897/2425048184638401450678617682989683492767892766876761372851401563449163864998598224579783679794912582768772963877)
(-4624342153798435367007727484325997952077426773322399/6455807132105718179909050640689, 3532089655334170220860025894992836453981159757222655901566529598906531134505385/16403095786221996086312963072392250958656516137)

Commentary

Specialization at T = 1547/3050 of the rank-16 fibration of X1008 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 66118T^2 + 260158T - 72323, p = -3001T - 11, q = 204774T^2 + 267483T - 2147712, D = -264840140T^3 - 585787782T^2 + 5921807000T - 7146678, E = 22(20828652T^4 - 235127344T^3 - 915628719T^2 - 443781740T - 35163849).

last edited by wgxli at 2026-10-02 04:45:55 UTC · history