curve #6164

y² + xy = x³ − x² − 1539009967811198296716430619075979x + 23518533230875197887555817767684022955520549793353

a-invariants: [1, -1, 0, -1539009967811198296716430619075979, 23518533230875197887555817767684022955520549793353]
c-invariants: [73872478454937518242388669715647001, -20320012711476170642422073504060163742820741301045501]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 791611698628038475107292196100229160812371724694342056749341780312711879429177670289845030
discriminant (Δ): -5654069735702168962636238480686719651232697780730188385563467742370712953695463640019724467568112500
Faltings height: 17.9465
naive height: 240.8869
primes of bad reduction: 2, 3, 5, 11, 53, 2521, 5984505821978086381294720657513510569250479881284929532648767545775099451848064369
regulator: 19718019324819264419452656.509594788402045997427597496227388151776183397670433539645655
submitted by: apkorte
submitted at: 2026-10-30 00:51:53 UTC
last updated: 2026-10-30 00:51:53 UTC

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

(21519195314131717, 604396397048041324040554)
(9176290627447627, 3188857738460679684988579)
(20665421767030198, 734582231515977027187651)
(15997193115544492, -1729895455689126275448971)
(324265802319990397/16, 51687019854569587594163281/64)
(-128239642206491657/4, 50537378065454860445290607/8)
(11376272254192867, -2735443133052834248953646)
(33514771905687142, -3095819043416387616155621)
(22667841124045132, -529093335750519589724171)
(3038739654657670153/289, -14322751787516149063468568998/4913)
(42056346211639423, 5760206708697235907738866)
(-5002360340709068, -5576023540192611358710761)
(-9389728989145952, 6094387209815325066144541)
(108146025493057363/4, 10344553208997844573072157/8)
(23555356011709702, 580006719084671753340079)
(23087909290961992, 541347063154523682296029)
(25658510054116262, 960387963489804170852859)
(55100982134020963/4, -17766931455724630331193583/8)
(-308772605681762879/64, -2842928049368724354342572653/512)
(24017060638495852, -639952553763367044701921)
(-1091142339830108, -5019612839431638278943371)
(-4255352266884506, -5476358894967829388682725)
(-20041433045029931, -6805341695536753865230550)
(32859053575746052, -2902863120410124218373131)

Commentary

Specialization at T = 50/43 of a rank-17 fibration of X1092 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -906180833179571257456587T^8 - 1930046192084997346024728T^7 + 6011648406192813483229776T^6 + 8355472943604505878941856T^5 - 9649478959250650200991080T^4 + 7450754699414011152778704T^3 - 27984481563110444272430064T^2 - 17758766835719522469182592T - 3317283523979050195432512, A6 = 332150243825525078123196022294739334T^12 + 1126061667323176712064227845083588744T^11 - 2713185092886283310031062058684874464T^10 - 12015892423153220327930309833774827360T^9 + 7554319779210673373159460328484862840T^8 + 58673243715970603365442249112691953136T^7 - 9754551868147305475783193074338006384T^6 - 149812543158867507723474722158888312896T^5 + 23811683678756345915421047541045697680T^4 + 99714459519208696131901088663865684480T^3 + 64761255203716406202557626472759415936T^2 + 18947601952381150031159658003937314816T + 2277468715827369848274981635668809984. It is a 2-neighbour of the fibration given in curve#728: 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 2026-10-30 00:51:53 UTC · history