curve #4267

y² + xy = x³ + x² − 4379495549234897017366703511701733527212x + 111585213931435810793140240671412708524278991975026752179604

a-invariants: [1, 1, 0, -4379495549234897017366703511701733527212, 111585213931435810793140240671412708524278991975026752179604]
c-invariants: [210215786363275056833601768561683209306201, -96409624836760540526849786337825143091229062330635737952974301]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 174164423262524400823734779124493200445680667881133146381966747877983549516873060814351057410
discriminant (Δ): -3031180607881395918727990838810988530528252705643095198533912996987108676033069476084758856941922656575825242518712500
Faltings height: 21.5164
naive height: 285.4474
primes of bad reduction: 2, 3, 5, 7, 13, 23, 41, 83, 137, 197, 199, 9931, 738643, 113737357693089399173, 59771131561917581425523, 3043301382609038760288509
regulator: 50434703246724614652393520.57532618660335327308413860702957644347421379819732226639379265
submitted by: apkorte
submitted at: 2026-10-15 23:48:47 UTC
last updated: 2026-10-15 23:48:47 UTC

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

(34387753032792608958, 40599560814594306229427654996)
(38834803367625820518, 8761370151871570915303170886)
(-45401540254074511817, -465655311285113394663609577304)
(39245657878937474473, -12491690944951924231509326279)
(42023100720743152813, 41899255645973096717899418041)
(38075066788914526698, -5784508446117612972728780294)
(4268514672276749946943/121, -41896164838877690131857340285549/1331)
(49764927272512310347/4, -1943598333573549243904701324607/8)
(38810497566553665864, -8561995885386226604827699142)
(11079579018889942751382/289, -28385642159194136562945440581292/4913)
(1895458537976506789462/49, -2599184883101323152012300405812/343)
(48667268157246151155, -117114046080035887415174606927)
(292476445850736421510, -4883630198968158578558532632108)
(41858051743336234733, -40092498074565265955391799879)
(41105139136600020718, 31906386979565847478717678846)
(19071519577852573158, -187078396543116684038377503404)
(-75546528611020948427, -106187853059531532529102478279)
(-52600862430709450167, 443183621450767935352605028921)
(38272664459386733833, 5650523601379774333085284921)
(400204080809931534622/9, 1864290343188499468963302133642/27)
(1586799850900662734776/25, 37357885249383856179289071737834/125)
(12711137735653318359352/289, -312561543734160543700659474738382/4913)
(38346640398488249658, -5801640051510814567063236004)
(-63659421119248285592, 363868337949293874269000692846)
(4504756476787142172943/121, -15761153074852032267030414123949/1331)

Commentary

Specialization at T = 602/177 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -527341140801402202800T^8 + 28147421355562296077760T^7 - 459551759375096200061280T^6 + 302351267241048825914400T^5 - 781968800173412856857040T^4 + 675358902068135754436560T^3 + 3773048587407798267814920T^2 - 1117382211836832683478780T - 4020150123856377627416667, A6 = 17121088738199440047220335446400T^12 - 754628062801204312746404692573440T^11 + 9968619820015289906033818763984256T^10 - 51608554478585574595758963618583680T^9 + 526057596644710242384434966047683840T^8 - 378806199877031919118656099183432000T^7 - 1773421996954369325651890262968908960T^6 + 1196920464887427116679365802787354320T^5 + 3649727616935479672514463019748893140T^4 - 2090084281847145605737007981579064600T^3 - 4861031233659870921990346669248956340T^2 + 1412045054473207996392438591189858420T + 3148792632832054337655276106728582390. It is a 2-neighbour of the fibration given 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 2026-10-15 23:48:47 UTC · history