curve #1919

y² + xy + y = x³ + x² − 62567866563741209951076219033072398452307404054845393938x + 190491265471364425314030862500734330699185709037646180838115059299563818726701936031

a-invariants: [1, 1, 1, -62567866563741209951076219033072398452307404054845393938, 190491265471364425314030862500734330699185709037646180838115059299563818726701936031]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/4ℤ
conductor (N): 1564270218869624840947724233813568092984047710847969648779404950 ☆ record for ℤ/4ℤ torsion, rank ≥ 13
discriminant (Δ): 4813836057093818801619023279105170848620819146585249722467951212550403148365457041430085193497457244019566117965687426235823978116085320214411090904408391680000000
Faltings height: 30.6122
naive height: 397.0411
primes of bad reduction: 2, 3, 5, 11, 19, 31, 43, 103, 113, 167, 307, 317, 439, 479, 503, 823, 11831, 13151, 22573, 24469, 26452134174795739
regulator: 17260732195257774.17575499450677976892658355062312588880924061905
submitted by: dujella
submitted at: 2026-09-25 19:07:47 UTC
last updated: 2026-09-25 19:07:47 UTC

Witness: 13 independent points log in to add more points →

(4558605665741491355288960045, 947294848874354296877229543522035810527)
(-7287328329039647624534080735, -509361519482732606832330160150157801204333)
(-6114466113616553353118054555, -586907727393166774178165578041114100760173)
(4162938925878945304973163545, 46572971665370560190189017803900962061427)
(-1260145601589984620133324055, -517044309291902107607923017734664705044173)
(6758529895986107600322149395, 276294675885672711822625183672118744105477)
(4564572468874153614771800345, 201728715671393249465445874353976583027)
(35922258661656712682968304890319/11449, 193955272921595967329755824908328738720948950629/1225043)
(97077179220453965996266175774345/7921, 792737671912824357948666003119556337114605221963/704969)
(80455314704368947610726303492565/22201, 352290034777633737341720978678255330978132766583/3307949)
(4302626795197794458885287961, 30624885865611638490727536491801094928243)
(-4677633058276326924348103386335/841, -14763483773060365677969959804737674309074580737/24389)
(4341864183441428105718170495, 26114649109698076841404659050696758181177)

Commentary

found by Elkies - Klagsbrun (2020)

last edited by dujella at 2026-09-25 19:07:47 UTC · history