curve #4595

y² = x³ + x² − 1075078433986621291499639542244317724233x + 18195083132515633277430966183757612134284743704318351981563

a-invariants: [0, 1, 0, -1075078433986621291499639542244317724233, 18195083132515633277430966183757612134284743704318351981563]
c-invariants: [51603764831357821991982698027727250763200, -15720551826493507152009977371754723815973914748697419616649600]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 9281497273884032385186528170081105432870619335479591431082212913885764058460952904515687647400
discriminant (Δ): -63493969483302358805700173285996889352501504303141641686348715559079738970866645922786893378351597345140172081706720000
Faltings height: 21.5024
naive height: 281.8201
primes of bad reduction: 2, 3, 5, 11, 17, 23, 29, 31, 89, 163, 997, 1823, 26723, 12950204567, 61287157423, 7153966458663330275159024012743280414303612447491
regulator: 1571625325451070369898098652.27545509665952170683573215208810855244107723267952200531840668
submitted by: apkorte
submitted at: 2026-10-18 18:29:50 UTC
last updated: 2026-10-18 18:29:50 UTC

Witness: 26 independent points log in to add more points →

(7328300453146984363, 103489827102602256847570479330)
(-103180569927148316923/4, 1356774310261299525288370599735/8)
(88895528754827607181/9, -2495107619553239833554599679998/27)
(87343593932236349903, 768524371607193462564460954830)
(-31990565948222476147, 140884433271559919652026943330)
(258449206192911940709, -4123555877093681502825041041626)
(68388801507707832203, 514322961555330363521577128670)
(199138979609784473333, 2775108428387395931452161144330)
(30405643573465861907, -116690809640921578920614623218)
(36480146018456994323, -165903229726263613056903494130)
(8802054882837427960187/169, -706755844483833015328102751743110/2197)
(26089883462230046624822, 4214129390748152100261669241185687)
(57798434090440602413, -386189190927185487665593803330)
(-27972998563509365347, 162418277228516705963775334830)
(-124267711006996424873/9, 4708143623324926118025847679590/27)
(23656939270662787823, 77470267335049958516122928370)
(-525578586137913860691517/28561, 860027642441747434249183457522365470/4826809)
(78477208631877988787, -645864798209414673479926004658)
(-1980324513066466997952487/55696, 1407709988280403352965584108852024645/13144256)
(41968100647429187841542/121, 8561014147238084588934992164076757/1331)
(-5773703192267331549097/169, 269513314467901068389032033271226/2197)
(-9777809863251923573, 166649880301853240137109881122)
(73883603455659781013/4, 544893994239027865371642446271/8)
(914657760276408779622254/44521, 649437151191045483451191264986300991/9393931)
(5948265194483548551976847/208849, -9863050989027549183604799657196608190/95443993)
(-25143722977897107697, 171261568023206228150840746830)

Commentary

Specialization at T = -250/51 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -15108348426986418946992T^8 - 165645727401183163968960T^7 - 386516497356749936802240T^6 + 804907699137855510269760T^5 - 423101056239642087896040T^4 + 2012740403434408357952040T^3 - 832891923170874739577340T^2 - 2371900038780270082074480T - 881248331824987560372900, A6 = 717503217218577271350459150984576T^12 + 11638864686052244686069138775512320T^11 + 58135061140664436934300824832212480T^10 + 62512031622436328327919258342862080T^9 - 268895690113423321891997914072476960T^8 - 870815838806588886732350721838471968T^7 + 2170770437820474859902147687346459920T^6 + 114853264658855014380401497670938080T^5 - 2228934495975748087928517866419014780T^4 - 844990471869666800949969469071688800T^3 + 1481780271644919349838123819347167336T^2 + 1307631772289943212553000319429645920T + 318018311456252316549507444881168100. It is a 2-neighbour of another fibration built here: 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 2026-10-18 18:29:51 UTC · history