curve #280

y² + xy + y = x³ + x² − 1475758023603928900220439411x + 21467548255862634426152509047650717315889

a-invariants: [1, 1, 1, -1475758023603928900220439411, 21467548255862634426152509047650717315889]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 20270237449228516491099532903714605797007023018097766862110470 ★ record for rank ≥ 19
naive height: 199.2905
Faltings height: 14.4941
discriminant (Δ): 6606173027470846981780960330759477153956670797167107093900274692986568514047180800
primes of bad reduction: 2, 3, 5, 7, 13, 19, 23, 37, 17714696401267, 97575958269840227, 265666313464398978169559
regulator: 9995598135680440074918719326.240258542580492471704347358452558664802178323944
submitted by: André Röhrig
submitted at: 2026-08-22 02:40:40 UTC
last updated: 2026-08-22 02:40:40 UTC

Witness: 19 independent points

(98995268132694178214628411/68549871865225, -78927998943145195686307744492878720711514/567557635860052409125)
(47098770058396364538867325971/2753937335320900, 5070396143240845693847335380447564293947209/144521040864850130573000)
(49307093845763299014062283/5798333968729, 1363239750082259292121371267196272889462/13962231641682276317)
(-330471590535102312485407525173/9623316094644004, -167944062712737155274923323105293145815415233/944032893157067015682008)
(6974098471304448209779203333/608344022366161, -1167663007088018800245402472360031958675158/15004580832717107951591)
(6389392077778454209647374739/472294732269649, 647437388656756137401976224749880624902146/10264071118777180557607)
(1769027674891492185949611034791/90155494104312025, -218506584522047430179373892476240633023442574/27070002561263859008516125)
(559245366469374984321654135819/30315317616573025, 120516412646780152908970807639945918762826582/5278289228831910190308625)
(2826452905856825450619239541800163/163614667455851416561, 70323247641860418390142544642886165300289021599858/2092827103035161296334015653559)
(-1377145755631412370158889837/62065830406969, 101733597390309866712741123841770531638018/488966218256387885203)
(4664077278932219285037464979/1324503364950529, 6156964239916742834452773967861128429291570/48203613874590920931583)
(-86978993454948821252355327501/4164772291292209, -55849982615745478178924029930401111592138206/268773775562840398548823)
(574047914207335863400029483261621/29770333469230287025, 2183780551293980740160520251883824380474400985366/162433480099235303248610827625)
(4562764035552125845634721363/230728474683001, -14584515857905502506650915387069907555962/3504707617587639756749)
(103962014312943685782436265787/5895488269818889, -13785016504620714256122518292086693972349606/452667869658760330165787)
(12198593116654114569358606378131/956277584677636225, -64232873526254245232983840702822635018991480674/935138557059829273710994625)
(6848483737973248227127540179/362055384229009, 123372998345530546471486642570215944957886/6889100423429678686777)
(308340221369694912986153391492003/20089285248110916289, -4441039642256212863027127724577999980935606185298/90042330716358137485184418337)
(3802550824058101422653449805454030027/191947574256973943811001, 219826828934460704864357881342649533082121426808806634/84095729470617512146111828688908499)

Commentary

A new curve from rational Mestre/Fermigier quartic-family search with canonical primitive root sextuple (0, 1075, 1394, 2291, 4186, 4824), at search parameter T = 2697/2. Rust implemented per-family Nagao sieve through p <= 1000, a native rescore through p <= 6000, and exact PARI/GP quartic and integral-x point searches. 19 submitted points were certified independent by the leaderboard's exact verifier. Found by André Röhrig on August 21, 2026 in Austin, TX.

last edited by André Röhrig at 2026-08-22 02:40:40 UTC · history