curve #4208

y² + xy = x³ − 144411785241843143757041787837713708785x + 664440312200785148892368165981682106414633675158209573225

a-invariants: [1, 0, 0, -144411785241843143757041787837713708785, 664440312200785148892368165981682106414633675158209573225]
c-invariants: [6931765691608470900338005816210258021681, -574076429741478368653403743945586046292750504061008458298921]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 863147432773276692280622020482356041379384990774069866239356534650083835203112312990133085330
discriminant (Δ): 2027353082094439742535445592333122824694671927678035010023665980654437494687808362129147158280051198221219840000000
Faltings height: 20.7723
naive height: 275.2108
primes of bad reduction: 2, 3, 5, 7, 11, 31, 61, 52452600344437098527, 82239645115100520642843359, 45807144898958519315103658828515673779361
regulator: 93370544687827192244336032.67233129867816260691267172826172787770717691283962017220191176
submitted by: apkorte
submitted at: 2026-10-14 00:18:47 UTC
last updated: 2026-10-14 00:18:47 UTC

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

(6518504828838445670, -261846355093603520235621085)
(6521403228909248522, 140112866095844676854878907)
(1100694534711209551130/169, -886055472937333979616345528745/2197)
(7444408190697983270, -1394016071811700548542094685)
(7906716430888042820, -4112768389225964087508394735)
(24670103763709899722, 110074327621000100661647710907)
(17128123375728143810, 56708550521470270934247072275)
(-13505709575677739830, -12301459745021616605812638085)
(527021466422190417970/81, -382855189220759159973544281965/729)
(6409576756331062940, 1464391947233102061475641545)
(245487414263358890, 25079948774216185597315575515)
(-10266831504664286710, -32632611864906668562138524485)
(9693830276783967170, -13246492212127289256759600085)
(5468080180246888970, -6187095185178737236577334085)
(5535507373108609970, -5887779878972538032777485885)
(7371554771905528970, 685481726803210986879065915)
(1237788308881452170, -22081339019364814896683910085)
(-5877265521707800630, 36196308426441596696451670715)
(2035455418969608170, 19466114492783500358718737915)
(7421226453928034282, 1203679855833546483876015611)
(6191054726626323242, 2770733706722989827175836251)
(7348228367868820820, 218758868182483881263853515)
(41903494902381703370, -261135117149177066561721630085)
(3758793486003192710, -13218640450619903093512612585)
(7368528353609485730, 643251986445568762955323115)

Commentary

Specialization at T = -6/1891 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -26975048530559638402192T^8 - 312966956619947949551760T^7 - 842243888708948190021348T^6 + 562167338991061514848920T^5 - 461942736201433628752680T^4 + 1431271445622423050083968T^3 + 603334350434130915358656T^2 - 986867998885806893128512T - 418274582761882236107952, A6 = 1451452485240104576307296743782976T^12 + 24581324346135835035118126720145920T^11 + 152392747348176550954751684612771616T^10 + 397210221029004103236543130531121600T^9 + 10829685735288357984444464355358836T^8 - 1893933432491868927718338035329167456T^7 + 859289861319319080984756807841283568T^6 + 1744608022036113544067265254248327200T^5 - 669413043714196611286442553109280544T^4 - 1068656424150865477675382650153889280T^3 + 65017381339639250150552216405922816T^2 + 368098108547413859417253532055609088T + 103576617923784908883415898755916160. It is a 2-neighbour of the fibration given in curve#4191 (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 2026-10-14 00:18:47 UTC · history