curve #4276

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y² + xy + y = x³ − 6549805135562652390648654106106130451x + 7967198525290173853615049509540291249526157411468931898

a-invariants

[1, 0, 1, -6549805135562652390648654106106130451, 7967198525290173853615049509540291249526157411468931898]

c-invariants

[314390646507007314751135397093094261625, -6883659525850710209994988746003322611717303099148798552525]

rank (lower bound)

≥ 25

torsion subgroup

trivial

conductor (N)

26976009553634069537301594434056775454262504651639756129924480206364431273292626306665719285050

discriminant (Δ)

-9438618116657891653374869872278198630193924973163805731183384978513795370133670941362839957752576147240677057500

Faltings height

20.1999

naive height

266.3530

primes of bad reduction

2, 3, 5, 11, 13, 43, 61, 73, 89, 101, 3948210856508020405740130399, 185062123304701313626594575556403822101399862455505701

regulator

162557287250877821015725031.0001738248656792595605405710156566359589904126634855783639839

submitted by

apkorte

submitted at

2026-10-16 06:05:46 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-16 06:08:11 UTC

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

(1663744331937481033, -1294340323609095414100531846)
(16728807864203049217, 67675727133322107851488122956)
(-161780403085327853, 3003763369278011652430300166)
(7884553925325979027, -21130016304508404921694374154)
(900276279036872197, 1673390537828720275528781516)
(5516883123394890283/4, -9983005886215585540125696917/8)
(4859442717289890772, -9533654485541827201839156334)
(413927854296865537, -2308023525536281038942758734)
(2311576914412639762, 2275625656873806000398062841)
(4152452492619897952, 7236684869078707078611075446)
(-2253381261564782318, -3359218890776460142918692364)
(15905995135817423932, -62673867252513812663804784694)
(-1774201043652296462, -3742066072079766472378608391)
(28959138611523305437, 155255732221698938485343075996)
(14238603967807636357, 52928356969229102261329356236)
(1789023257053659688, -1405489305218087922298571698)
(-2951607003313958933, -1259085593637378926406815974)
(469387041416085768922/49, 9847488797733619596975542335007/343)
(707367508053490743451/625, 22118978529318758228172988723874/15625)
(1075444440949620121, -1472102353051999192548147958)
(398303906231874817, -2328427554779914197849804724)
(-287989905561195623, 3135217843844246382216737546)
(481229081510796793, -2219613825695209590409511908)
(4878890180961028627, 9599292657403598235199591046)
(-1029446068433577764, 3690380024459610290701808042)

Commentary

Specialization at T = -30/1871 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -43748288271748043306667T^8 + 231005568889113088943880T^7 - 41811485832351502958160T^6 - 845807620575605275473120T^5 - 1273696848822391714464840T^4 - 2479538071066771770038640T^3 - 723198327599392266394800T^2 + 1473487148401141809248640T - 304202213366895598996800, A6 = 3496542619563277295766359856805926T^12 - 28800862572418754868158883533967960T^11 + 61983572330491650413223232787176320T^10 + 92074250730656596945854018024980640T^9 - 554403246688053294582105340433199720T^8 + 21665370800800153534189468438161072T^7 + 2696996106708736152461893481712643920T^6 + 2003709311601233451889158133572553920T^5 - 1216187330113512252280500336077191920T^4 - 278472048982945948218206647760751360T^3 + 596780792582732967834659478701644416T^2 - 397452607403627859574187730447221760T + 82758424296652691415712452035078400. 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 2026-10-16 06:05:46 UTC · history