curve #4171

y² + xy = x³ − 34037549680835403668306569395011044262150x + 2421919497178141543191669805725453957161094973626571613138532

a-invariants: [1, 0, 0, -34037549680835403668306569395011044262150, 2421919497178141543191669805725453957161094973626571613138532]
c-invariants: [1633802384680099376078715330960530124583201, -2092538445561914293320053415723812368051304130209798668938566449]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 1639273270253829061085618912592117264758680112319811738008352311590419724053396328877724528890
discriminant (Δ): -10180395461560240461826647859544283347335567998835505161890560874626445722868962817869037102219024785187411434741760000000
Faltings height: 22.0998
naive height: 291.6025
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 67, 482971, 4046766271351, 82010967778567575483026658540263840590868971744475882772604877623
regulator: 51318897545768907285166479.33272986114452805547995859333712934938264946342400439520966357
submitted by: apkorte
submitted at: 2026-10-11 22:17:25 UTC
last updated: 2026-10-11 22:17:25 UTC

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

(106798739081934637804, 69967101740219665272213053098)
(112113535405647113404, -122696222084389444677200283302)
(100458810644316502444, 127966038683592723231404557738)
(-133447415634902951516, -2141887618238554610752743203822)
(53467271186414303284, -868834915658222250415826137022)
(108509684195127496204, -78402170690530526307756671702)
(128682297119540629804, 415640186696685132348547421098)
(69457495310749684204, -626772188804936431336486876502)
(112964249695617464782, -135724656328502531338643061500)
(-132502898557484424596, 2146074951680017696836241905898)
(-111013139693633813696, -2198276146095086187705409427402)
(104035341817632563404, -82598683711117349351236730102)
(104474777083092113764, -78702506837941720975395663542)
(107692483567124994604, 72891926173998698193739843498)
(105406960707193840684, 72541695210355601114501566378)
(148417399839682471564, 799657728248519305540154991178)
(113305545094618214404, -141101274559541996833101237302)
(120909607814021703964, 272114651467208537629083873658)
(91391646683396885164, -272973251652646268885913832022)
(-57796346733064544033/4, 13648652690077692559845370158713/8)
(47339156076579598804, 957443191758136383587091416098)
(3809731822234704103804, 234877438505009286561981055781098)
(126109533771144661804, -367501547124580043453155746902)
(92518318565833658872, -254452481699998030268580612962)
(-160556876143469477606, -1935966175166728730009591656082)

Commentary

Specialization at T = 2748/203 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -8601190111439927121627T^8 + 70553802194275661608968T^7 - 40117514965744884873804T^6 - 394125757357369375667736T^5 - 727539205447352323463130T^4 - 3371639403550126915091304T^3 - 358500560703347306973924T^2 + 5575527756032095197563112T - 2574982297007819942866587, A6 = 311073935845060124132541860022390T^12 - 4299424530738788428378210804267800T^11 + 17883941445167312163435648485037300T^10 + 19652059401694206221665695004579800T^9 - 317465301013995782557083261357635310T^8 + 112305603376549774936723074091082640T^7 + 2598376408341179206723494257772920880T^6 + 911887687830002593185098844364676160T^5 - 4509956533653094383564183143089914510T^4 + 1581320920924061257174062701329500840T^3 + 3668404075692271898587012925610059196T^2 - 4676936909330964399995425467736251576T + 1568898839279682527117596188509969334. 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 2026-10-11 22:17:25 UTC · history