curve #5963

y² + xy = x³ − 27091306445851611235147121631509215x + 1726934946620741304919044043558537680283616709134681

a-invariants: [1, 0, 0, -27091306445851611235147121631509215, 1726934946620741304919044043558537680283616709134681]
c-invariants: [1300382709400877339287061838312442321, -1492071793880320489400628117735892564695637594161027865]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 3160766107543915573005279846610883009909959037472309770941987988415496938
discriminant (Δ): -15820214210325146503563031719148910017470260514683647213306318572090856697650778255460548677421764313088
Faltings height: 18.6344
naive height: 249.4795
primes of bad reduction: 2, 3, 7, 13, 17, 19, 23, 29, 43, 47, 71, 120629968427, 197027661082697, 414677700574817585493790585579583
regulator: 120886865519781707273.0417259857074429957884243159855390190777530136732005995665588
submitted by: apkorte
submitted at: 2026-10-27 01:04:09 UTC
last updated: 2026-10-27 01:04:09 UTC

Witness: 22 independent points log in to add more points →

(81643843220607878, 7701385243976555829699533)
(101643282171296894, -4837215847480877026800739)
(-35356400047183642, 51386650642846370214697229)
(515058096288668870, -352718647626762044275161619)
(85342364315464466, 6039410375745716373486341)
(103228892756525018, -5509768625434944190624591)
(-131534785002107914, -54905812287373264862154979)
(41964424074460838, 25767494343513655719235469)
(952494537660063286/9, 181472456655144737660278687/27)
(109183972991906750, 8402136492792606690940709)
(863170078341056150, -788326239347530150831313491)
(95917920266977382, -3295720650117131567717491)
(93977618032683686, -3308978365726782093910579)
(60183463908915830, -17733407990887459027169299)
(99068738496627382, 3918534684920344216032109)
(20600311197748646, 34315999396219234357055693)
(302966747779777382, -146041567471922190092501491)
(74914429019128454, 10855313383034825767511981)
(187271769857927510, 56756083775134321246966829)
(1290494734014916640, 1454622134004291203004165371)
(108856153469109469262/4489, 9903478050949342136512208240519/300763)
(105133896811533254, 6385891985915076988857581)

Commentary

Specialization at T = -159/26 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -1372222876353130936512T^8 + 45268756700642132762880T^7 - 453891857520504970746000T^6 - 18572366977872800377824T^5 - 1977700547530427488113780T^4 - 1507927812614917602109344T^3 + 3528535081023240424639497T^2 + 739749873585389019295698T - 1838633851571299960339827, A6 = 20363691452208588844425836248320T^12 - 988378436168557724966091861408768T^11 + 15965030664200057535359388315912960T^10 - 108360266161421346549906270794042880T^9 + 467598254108283083435022152559186336T^8 + 1192549643051435746625317280337017856T^7 - 1110484823304716478646019257293024192T^6 - 2091654238173550372318136090501079360T^5 + 2721622409681114022374952201370443921T^4 + 1932785120958781428078049659343305420T^3 - 2684653676111216975923215434658050073T^2 - 583595531728434724891836911712239106T + 959603028788038989675020832528690546. It is a 2-neighbour of another fibration built here (itself a 2-neighbour of the fibration 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 2026-10-27 01:04:09 UTC · history