curve #748

y² = x³ + x² − 494024163014270917031562635183457523397345x + 129933483086272039575669571305782391919405258982723205422209443

a-invariants

[0, 1, 0, -494024163014270917031562635183457523397345, 129933483086272039575669571305782391919405258982723205422209443]

rank (lower bound)

≥ 25

torsion subgroup

trivial

conductor (N)

13713792420445736434908008709350792188822397906478024811690216268911099779951692892301792169222456244028191480

naive height

299.6238

Faltings height

22.8801

discriminant (Δ)

423243655651351406628133403780642405167779882423177982068223504595709799732797179183781464989125500213729402499610463124000000

primes of bad reduction

2, 3, 5, 11, 13, 67, 103, 823, 3121, 4483, 252702087248153, 614730522279708207905232553732673, 4979995603879699209645396858755060879856491

regulator

34924277214770449210447732858.87680756907389055736378006454677912804015163376072044821873

submitted by

Valery Asiryan

submitted at

2026-09-16 19:04:16 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-16 19:31:42 UTC

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

(-634800275344802708969, -13701598765641475899165122947650)
(-1673889334475252691417304959/7134241, -290826074094445767389994887797187894301390/19055557711)
(283661018923144301131, -3552810848130721151097824693850)
(1624143547980205725434821/5476, -1248976366645999377078410979023552295/405224)
(339822793065031275331, -1138151496453822133623609303750)
(341645925165438796078, -1014748942607587592370144376263)
(401201768902315939680661/1156, -20835217203491453459405734332640575/39304)
(347882403548912521531, -415514235089480502707433538650)
(348431535111150882031, -317889864916523328189303439650)
(4157670336946940240779/9, -14790739240337157165189027198050/27)
(462045441482417359831, -558658286222692422378463458750)
(469562350589565660031, -1221300770615668989263632413150)
(495769569652291373131, -2620168454773929565366708354650)
(31444067587448995387519/49, -3013100373953779874214628003374450/343)
(9900725019625503681904129294/10936249, -745298215751319965685455426031177395596125/36166175443)
(245812914185850575333374/225, -100921979472888703622431753304469793/3375)
(1453293242382863649031, -49813811790886182140397947443650)
(69045540120717626250714023182/37613689, -16965130166050383596777917869814364040728259/230684754637)
(4497335474525859292591, -298113010440527248555029843028770)
(6745673506463149626931, -551138171568026520692225177050950)
(649209924463259499161041/16, -523013596859165620595108356483921215/64)
(59523242892222547484761, -14521107134264297384940766273574010)
(225183098030987346277486, -106856654135044580121036378188671335)
(16481463156809408638646546572561/60466176, -66910180804393672796126121188467286965199919865/470184984576)
(309629198586173493764286, -172290664100435436137639096244448815)

Commentary

Specialization T=-265/221 of the family identified by wgxli in the commentary to curve #302. The points were obtained by specializing reconstructed generic sections and running minimized quartic searches. The 25 listed points are certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 19:04:16 UTC · history