curve #4230

y² = x³ − 23611820850355718516136424825182012x + 2139078598949842081379814056798536764899142679364016

a-invariants: [0, 0, 0, -23611820850355718516136424825182012, 2139078598949842081379814056798536764899142679364016]
c-invariants: [1133367400817074488774548391608736576, -1848163909492663558312159345073935764872859274970509824]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 146954943734085658569266296693767955932028661605823373833844075363529407736700157718105920
discriminant (Δ): -1134186837575672884356463499602074052823250021137983885342779082105214743258517728044807046078985472000000
Faltings height: 18.8568
naive height: 249.9076
primes of bad reduction: 2, 3, 5, 7, 23, 47, 59, 613, 3555869869037, 4092513265733, 1423564104240445042058874623429533002566377872601729
regulator: 10472542313942194830406339.724476477756222552708539660971498597598694532586531095426645
submitted by: apkorte
submitted at: 2026-10-14 19:28:16 UTC
last updated: 2026-10-14 19:28:16 UTC

Witness: 24 independent points log in to add more points →

(55654725527237112, -31580956840366245941288940)
(2156886165493000602, 3159966162746193304291128600)
(57018687327565136472/361, 332501009406393850994417155500/6859)
(330668107715114433/4, -219394451618660768807417025/8)
(107441040213303597, -29025057092797715706966405)
(277555554845640312, 130259578217726502445990740)
(13156277060953477, -42786821927946676793558975)
(189117913367507652, 66615111979824706880013600)
(105942109278706746, 28751663938797880169825400)
(-86606930826612648, -59450888059781730705104100)
(346182444126924282, 188288012753807956263604680)
(-734251494639419367/4, -135793005386916128715652125/8)
(56784457447927002, -31327211873307098338233000)
(-98831223635709198, 59222620863337388310751800)
(296931712773607992, 145972554621873046762408140)
(-1373434231892387809958/7921, 22517310939584942886530849448200/704969)
(-46834958971168008, -56055366830215811466547860)
(7345784352712582789368/24649, 568362194414659154006630121410940/3869893)
(-180582316915812648, -22675347909497643234779100)
(10516987621168796073/4, -34050204819350837691353944395/8)
(-1598509038201786572/9, -729430360192367437357404280/27)
(235022699371382002, 97833536209474203500243000)
(151981981298954712, 45398970471534538557941340)
(-10409431572388398, -48823522048888318597895400)

Commentary

Specialization at T = 103/40 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -10669947921935414766592T^8 + 11909462495223346810112T^7 + 29290850738695015599056T^6 + 749826724411797003953776T^5 - 2800296645908250177432680T^4 + 5291011042236720751525024T^3 - 9053743075112346775820704T^2 - 1373686135035664689321352T - 153062929487822447776267, A6 = 424084459520579325699737587892224T^12 - 702971076900016054991835746922496T^11 - 1428460164091780099364248773693184T^10 - 45531551300070380060674898437602560T^9 + 178145138683306119166957105019549520T^8 - 224307849011465835658368095576866752T^7 + 1890363249575504635685922631864051728T^6 - 8402714629932209203714457518206796848T^5 + 10932033646871752694840038266436614840T^4 - 1045636958253350245306227479236779680T^3 + 3911537622199302231442442603656239344T^2 + 397823992391227625022128206594583384T + 16543056563137220502197778370669126. It is a 2-neighbour of the fibration given in curve#4205 (itself a 2-neighbour of the fibration 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 2026-10-14 19:28:17 UTC · history