curve #615

y² + xy = x³ − 23820614438915315378949854267269436396860373022138230x + 289224041721777817069595928142668314551014692398596976042862388473790867636900

a-invariants

[1, 0, 0, -23820614438915315378949854267269436396860373022138230, 289224041721777817069595928142668314551014692398596976042862388473790867636900]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

18953687733083223999142325594019759588962896426225863549465819845748137281442267116736751293352927963809555082278665495077084083130

naive height

373.4208

Faltings height

29.1841

discriminant (Δ)

828908268243177215456565575627703375854124622299768004485259797664586721143980604759965393068114386891877176310319576170645244768536922035101806458011136000000

primes of bad reduction

2, 3, 5, 7, 13, 23, 29, 53, 149, 709, 104804053, 1333322641706513829179052506975314577, 13304039783244056077097891497661130559978681974428774803618115435894522011

regulator

3437203961467798821467356243105651.378640706816933147798919920761466972317073052903314084629222

submitted by

Mundoamundo

submitted at

2026-09-06 22:22:55 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-11 20:49:20 UTC

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

(-118249068868718784740476640, 1205210122042022997203218864720096032070)
(-65568178309260503842890650, -1252680542680546932872725149457804273220)
(1331631721352041102281221650, 48268785169187725563356218812101461291870)
(-63987393846694524118166330, -1245573744664262050253633315048367759140)
(11916205923279528534139660, -84052055246864137806476062666034575430)
(11374543271878722997384780, 140524279400668174908908542860114254650)
(11864543999314444109259310, 90958532764585116639427455506897201620)
(190288502750113194374896860, -1626876547464681830231826016711680503430)
(-32947534961315208940121414, 1018964523262872445161048704802862949512)
(-159777648933077693652439940, -127587831108996684582545454583581146630)
(3965367282576872686991853244/25, 88516543994823179658092931920826452717762/125)
(175432673043709857090433260, -1228631424463720502408686234140110005830)
(674229398502747521805257980, -17050613672723297690094904722629552605430)
(-957080630140784241879551915/16, 78385740345903570839108760235013836429855/64)
(278355601213660380669895660, -3902067009548920515731379651029693995430)
(-103539929767942169899975940, -1282812225021892938271959720135171943430)
(-54003548505840159141393840, -1190851252106480355742353722155606917930)
(10429910741128583985950427760, 1065058317435586184881751659910426338132570)
(-4240823599287528328955222498420/73441, 24148484479737097834487142702784947258043903270/19902511)
(-658437414247106928470611806892100/5841889, -17534970601907936945204184507600998689999101033190/14119845713)
(-15264580204302596976695452156310/208849, 122220791573316886978481779773756329364817800910/95443993)
(-114726904727435974317444020, 1229643749163318226208295726725497504570)
(-44670849989955536764240640, 1124353650814011390207580120146637036570)
(387322555556950788554546860, 7012033448176869629710810628849506876570)
(54592963988074244793983770660/361, 2615500704314580418278697646347231097713630/6859)
(71123865449354547162902713540/289, -15008870381287561165766277392088564965222590/4913)
(-22117119174178643716673162716340/908209, 800256785973697720376543609620171672814963907530/865523177)
(253696302094708560140194780, 3251826621433321072474151610647517494650)

Commentary

Fiber t = -1707/3023 of a rank-17 elliptic fibration of Elkies' K3 surface (arXiv:2608.25406). Found by a Mestre-Nagao sieve over t = a/b re-scored at 2^20, then an iterated 2-covering point search (fake 2-descent over 3000 cosets of the growing known lattice at level 0, minimal reduced quartic models, ratpoints). Basis LLL-reduced with respect to the canonical height. The 28 points are certified independent by the F2-rank of their images under the 2-descent map at the good primes.

last edited by Mundoamundo at 2026-09-06 22:22:55 UTC · history