curve #4609

y² + xy = x³ − 947560225636838217135995520157246970675510690x + 10994069953961002874896208436141818033594827970406515864768400865092

a-invariants: [1, 0, 0, -947560225636838217135995520157246970675510690, 10994069953961002874896208436141818033594827970406515864768400865092]
c-invariants: [45482890830568234422527784967547854592424513121, -9498876440222306483910392313162776633377565158108681028941786984209169]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 123002866817579179599597934355591467598164984173082841396259480229356238688609053812425123766047888102610
discriminant (Δ): 2234664670784224944273556889649578016609425270762988365476466306176039961087386083196666358984895879922503394600649747919679219123200000
Faltings height: 24.7562
naive height: 322.3010
primes of bad reduction: 2, 3, 5, 7, 11, 13, 29, 31, 67, 89, 101, 137, 617, 6884381724913004470688326918874802467058900576346123822298751942151953125883446533663
regulator: 28307660299460390964986699967.792453733483416090956139307209414511254917392840727289978325910
submitted by: apkorte
submitted at: 2026-10-19 02:21:46 UTC
last updated: 2026-10-19 02:21:46 UTC

Witness: 27 independent points log in to add more points →

(21510193575308197659394, -751254763720642858045069548552452)
(-40758957281564668408067036/4761, 1412156863482537370227471254964035565142/328509)
(2632839170409954664429348/169, -251932865964692067041966250090585050/2197)
(209680846110508054931572, -95032357222855455674821272709695170)
(131130711391701196276024, 46277067267435648085467966884042638)
(15250641415246042415104, -300323799673683673043205381607022)
(15060702431937747335674, 373212459422939193249230769530338)
(19830423986501814418084, 42168658595905993947419879491918)
(3073248885588648087244, -2847983136891225139209997454801402)
(28283987531362018306774, 2611517649360275314196274355418338)
(-8281090676859593214776, 4274694632081804937987990236965738)
(-8012429581546264468748, -4251110366324435246161355385248690)
(216434217943705453292524, 99722136408425683659228939852734638)
(15403589525939261957464, -230346821950725110170028233325042)
(785128814880537821842204/121, 3011373160595593325815382899022745578/1331)
(-146761607400384533508644/9, 126956235105983397219800148757951186/27)
(352736630737849716959524, 208723143816982933168982295834789838)
(22845881701355063452024, -1127062843230122798334127001109362)
(-13135201328113229617196, -4601543420074895245806232325150882)
(-113916197429807544953489/4, -30861313877803909106733969993188731/8)
(-2099173204157220103098956/361, -27698307427953995901180457964872959398/6859)
(22464901331417804296324, 1022068974017329573345048038554638)
(80020056460981166803143244/3481, 239449964324538869706429826455134104202/205379)
(22105043856129904747924, -921663011140400907178403511224162)
(-34284093889001814615926, -1784046189586091372958050576237612)
(4679517916771780061082776596/299209, 190385440621246738596803071160736268234/163667323)
(14408108524339087777324, -576671814224786257163185344609362)

Commentary

Specialization at T = 5411/3949 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -58701756188828782740672T^8 - 421589935203760423753344T^7 - 677464368389618755113936T^6 + 669015607734413855170992T^5 + 1311645217705022688331320T^4 + 1050713078894545198928832T^3 - 1636359720229735222434336T^2 - 2517746756775352549692984T - 810344655390742277047947, A6 = 5480627817022307258018449212308736T^12 + 59387395680501866438327388746270208T^11 + 202192728222021747784840349845289856T^10 + 81426836279267976104124948661760640T^9 - 731444121000417496333010776856855280T^8 - 504969396495943282941884145271012800T^7 + 1906585932510725663650827845073118320T^6 + 512612498829370479665585111044169040T^5 - 2456755728368285626110680543972723880T^4 - 642737286070578650255592249597744000T^3 + 1935352237260464424985879727209027440T^2 + 1451023268234470568001166626460852200T + 303174450039421684871841546202620870. It is a 2-neighbour of another fibration built here: 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 2026-10-19 02:21:47 UTC · history