curve #4205

y² + xy = x³ − 111935435739683533234874004071486989443445x + 31998009985459485675010147920447916054753653584858326348507937

a-invariants: [1, 0, 0, -111935435739683533234874004071486989443445, 31998009985459485675010147920447916054753653584858326348507937]
c-invariants: [5372900915504809595273952195431375493285361, -27646280627436995623216827154640256685700067625610741028350785609]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 979409504419896215116182467439254200955571188757207790219066305425460761435416461348377034210
discriminant (Δ): -352552999829263681991029569532859586844198362034699175226580886764938405380171285356667243171936320073457306159450267792000000
Faltings height: 22.7788
naive height: 296.7647
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 29, 103, 137, 293, 41890943, 3466096787, 9396217511, 14868139758072059, 53187089596512271, 10961777702801604793
regulator: 36283142742820593232603135.64812226685178757742818290974748363391051493102652911109328068
submitted by: apkorte
submitted at: 2026-10-13 06:57:24 UTC
last updated: 2026-10-13 06:57:24 UTC

Witness: 25 independent points log in to add more points →

(-430974524505103086646, -436504753163215553526351291217)
(-431282026217274092726, -231277396558468586995386847697)
(-589113183202922650369/4, 53837724732704877234360722858869/8)
(10954253034071526314, 5547355708952212364823485782343)
(212625146290836057914, 4220235551257346598160495822943)
(818583279697237513594, -22110736313392792819241191313177)
(-131293776961743791806, 6665672258516273300017730546423)
(1664752028775100357306/9, 113341597271154182925148689356501/27)
(482236745969274185024, 9495466500517104002451683117423)
(82025587408214429114, -4834079480913535803044474669857)
(966089402129518336714, -28732166110245761971353472434257)
(-267123365697887011486, -6545076109203927028790502258457)
(155573188466321919428, -4283596370597718671452804099819)
(232969447323236766914, 4308692958885999753855388131143)
(-352314543590275237606, -5263388623451814963534649724377)
(161441249820478607114, -4258485808079276265040936676857)
(321017918295087042164, 5398743343171957578111486434693)
(8642778609811336549694, 802906953069629247915774660912383)
(-423027622216463997598, -1909972772688973221766525638217)
(260779767920004388814, 4532342077558886402747012669843)
(451891285588895981678, -8584529904885921368925812210509)
(47536442803839103964, -5175366294618306856583885510707)
(381710865468695045264, -6699815423815473649631545474207)
(-120653594915480301826, 6614154015446921371465926293123)
(-369918552796879756186, 4773410456502860229859641728843)

Commentary

Specialization at T = 1249/283 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -25474223654459729695152T^8 + 93424157542535492313792T^7 + 249549530991798533589216T^6 - 697407234931421362933344T^5 - 624235669206688152877680T^4 - 597548430366277034810016T^3 - 3503195429735821784150304T^2 + 3443481263994739372563948T - 932157668798019061834527, A6 = 1558176592694379366370572668882304T^12 - 8556444096428535116807109681029376T^11 - 10909694949700311770505743225764224T^10 + 132365209573836556400929013899470720T^9 - 65939622169833360804191934552439680T^8 - 921173694379955547822888188003154432T^7 + 724370040233725908757397970764697408T^6 + 3685802165670854989077288042679139952T^5 - 921815011359411216952545051496976700T^4 - 3541405794655636698734958615621659160T^3 + 4046594122834363561792364624080217124T^2 - 1844492709789748579332256815007671156T + 328955182757185562525000064586958946. It is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-13 06:57:24 UTC · history