curve #4128

y² = x³ + x² − 301951420222973517294997043878833201620x + 1997194382118804217053599202855012237294947436366323660064

a-invariants: [0, 1, 0, -301951420222973517294997043878833201620, 1997194382118804217053599202855012237294947436366323660064]
c-invariants: [14493668170702728830159858106183993677776, -1725575946150646843621271720290946946003793733657607604361920]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 10656469625163555146021797703903917216964830176148814097509882394597808021211006430503491817684904
discriminant (Δ): 38785067411977966069465872963985384735649013711074982813679012184376449591768063923146928263147940662814187186548992
Faltings height: 20.9882
naive height: 277.4236
primes of bad reduction: 2, 3, 7, 23, 47, 67, 101, 12853, 26790697, 8177556336688786843, 3079428567550772047311716799250878820160039107079231644553
regulator: 2093866981009975950392322688.713185426089533683401521698886856897273125286675800659043152
submitted by: apkorte
submitted at: 2026-10-10 11:08:55 UTC
last updated: 2026-10-10 11:08:55 UTC

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

(12373096131245241190, 12464394197735746877679644058)
(4834735801552327119, -25501948859743985114695541598)
(-18322110431811928580, -37132924893150156615182254008)
(2043964606731039555082/121, 55148036460143997566707921851294/1331)
(7418233778886799030, -12863708219055395237492006442)
(112209166490286271555/9, -352632493398929460221282250284/27)
(5330912177854231198, 23216698110317858789506321170)
(-19765830314921989976, 15596193574167029336519348328)
(1003534368289368131374/25, 29196920114535898664931486233718/125)
(12070781141767584622, -10543388446674480930181551966)
(24050808991954710370, 92989259544391818340734624258)
(13707649760276152374, -20828173831967902642049076822)
(15311325375170257294, 31039560235073920971069848898)
(-14848365482863593971/4, -443037835047582225164187894987/8)
(6388673136365875951, -18135050926719781537033715514)
(36297234726783427261/4, -16781887707150046785160301571/8)
(1335197416329173627350/169, -22485845999917196142893645537826/2197)
(48594584791715063482306/11881, -37322986628447109873453716558109594/1295029)
(2165681813175625193387845/198916, 46326161845681226762408620131315663/88716536)
(1464186714046970099404/121, 14288369997406506767979758063568/1331)
(184915324103204607655/841, -1071677548893535144504769980893438/24389)
(9103064634938305690, 1686324673785740947996214058)
(639230899068810771598, -16155774340729143468197149492290)
(11816675531633432706, 8896065115304813665033833186)
(10953038131723886059, -1983552242072890057685164188)

Commentary

Specialization at T = 171/170 of the rank-17 fibration of X1092 given in curve#729. Found 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 2026-10-10 11:08:55 UTC · history