curve #4192

y² + xy = x³ + x² − 1562296098883324604024432557245131642x + 546524047811763443622405545086073864779426011919003944

a-invariants: [1, 1, 0, -1562296098883324604024432557245131642, 546524047811763443622405545086073864779426011919003944]
c-invariants: [74990212746399580993172762747766318841, -472196777309363615852184986552364676618219794906266798861]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 1313579416141730104084274184163432663820045517198712103598711300093553255800439215630 ★ record for rank ≥ 24
discriminant (Δ): 115011611774323845477465906815080040282744050265786503715848172026329812188124053537452991083024785900798437500
Faltings height: 19.8265
naive height: 261.6313
primes of bad reduction: 2, 3, 5, 7, 13, 19, 23, 29, 71, 73, 79, 83576515082527, 85345050910342772012547568624894287280618106178000028121
regulator: 412061463646648447184690.94535341252139133787971596615793707006250202139859542907837032
submitted by: apkorte
submitted at: 2026-10-12 03:55:50 UTC
last updated: 2026-10-12 03:55:50 UTC

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

(1512417416644476893, -1281871117517527706963755009)
(1456427226005543143, 1166403944317428806342551866)
(-12356377372728169463/9, 8688699787119573646086076757/27)
(-1304188956732576182, -604770078207892273850145384)
(-1097106665006657482, 969538403748371958951978116)
(-246560383708345107, -957463037777901726658275009)
(-442586230125082857, 1072977101499285579023977866)
(-750339466352757857, 1138564166392752630469480991)
(-839818308704980327, -1125277492566638655849584209)
(384601572974529643, -50518793506233183228410634)
(17650240655034052403327/289, 2344420428186604634415544550380783/4913)
(-68262423097300297, -807992622900683073085399969)
(1297215634979586418, -838333231004115070883722384)
(1269096005052811637327/289, -43492600709040145957659848327342/4913)
(-3566463324532889491397/2601, -44129621507620689725674337258294/132651)
(-1556360155753425844655/5184, -371096001325524539271587995088389/373248)
(-1320706243082169757, -553350349292115134802367009)
(362410479197279063, 167125861156739379287967071)
(194400110025824393, 500160141411509725017220616)
(1010848208068036665, 13371143769475303209445272)
(21016124361264551833/81, 290357776244442223141839228439/729)
(1099532101737011038, -397531868881036904397696634)
(-1875868522432687829/9, 25083833393710394868412283141/27)
(336085733160637868, -243763652517759274854503884)

Commentary

Specialization at T = -5938/5319 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -4437370829345071269937785502500T^8 + 24731294533954602783634178920800T^7 - 24905343694917325040325875752236T^6 - 73371957550294374341624859203376T^5 + 11388134363003403434011617130260T^4 - 31974960227112637988116696606080T^3 - 53087557802286047432098781752884T^2 + 33902036804480905040632762295376T - 4900264269756058995659292474432, A6 = 3583088599917473486225042501788936057800562500T^12 - 29952862178447015859986519449513030145545410000T^11 + 74261855654700672409867052055125389734923941200T^10 + 34915854636953526118546131005709436129082490240T^9 - 320350491450370659682737733420310606418445799256T^8 + 104052954099599046113301816294931102834227932832T^7 + 846552244341643086726187489356042742190493998112T^6 + 378044570078449717959306675143150641014761409024T^5 - 387170892915283488485440730375460569591982158796T^4 - 122143831120324014405241873122119111177801883600T^3 + 159312688808409642522129729739582813691214920688T^2 - 45075775528724488962464647393974897773506063488T + 4205873318402097235409142015354269085411308544. 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 2026-10-12 03:55:50 UTC · history