curve #3612

y² = x³ + x² − 120872627034450933672056108x + 502292085827363419331748909982151898288

a-invariants: [0, 1, 0, -120872627034450933672056108, 502292085827363419331748909982151898288]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 377744506397500932049149835304360449085422198510087093787400
discriminant (Δ): 4029775546283096723137009505338922627499213359242882233035478042425962356000000
Faltings height: 13.8733
naive height: 191.7839
primes of bad reduction: 2, 3, 5, 7, 11, 17, 31, 59, 495950017, 47477125039, 11167892699404202375307704232253
regulator: 136370476561946243122803.6908268312234518613933011794846798383062542641242533
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:19:13 UTC
last updated: 2026-09-29 19:19:13 UTC

Witness: 19 independent points log in to add more points →

(4843116072103, 5521861620285006150)
(8741315676994258, 817268693594497006196250)
(-11407698690033954662/900601, 860901498296491425455003850/854670349)
(-211883264194251761/19321, 60588614747677471488459594/2685619)
(23207943709469908/6241, 5033680315701532018547100/493039)
(49560427024689982/5329, 5256087038107639203704850/389017)
(140554790083657693/24964, 1871073275683010996413971/3944312)
(473343312296232148/434281, 5518684383817872362650100700/286191179)
(1075321192733060722/12769, 1106012703616323520277233050/1442897)
(1363210227868129918/192721, 93358125995544907788964050/84604519)
(18454970299364249737/2534464, 11574530287760793923853977925/4034866688)
(18706546222818044116/2569609, 11769798461112706835366154972/4119083227)
(27067521920469491682/5442889, 62438943659282049937814894850/12698260037)
(31527790992358952008/2563201, 121491261846973016935799316300/4103684801)
(32174774174414738452/6265009, 64628403178558830522115436100/15681317527)
(194074867633505639911/172225, 2703543447949752794907258965766/71473375)
(253345157154430352212/35081929, 509535684701799146585627984100/207790265467)
(5321615386491914066932/679279969, 106479912862910242002386573282700/17704073832047)
(8995983635450338643502/1668967609, 185155286813754542478557138393850/68182333730477)

Commentary

Specialization at t = 5/2 of a rank-17 elliptic K3 family (member P = -3G + theta of C546 in https://gist.github.com/7fff-zip/58764130a98380def8d555e30ba81344, in reduced coordinates y^2 = x^3 - 27*c4(t)*x - 54*c6(t), c4 = 113915556676586218390225 + 117014247072745494659640*t - 714542170231101375740040*t^2 + 59737931954947719760560*t^3 + 1321595301880580949151920*t^4 - 1928468740555818605289600*t^5 + 1536514504542469295299200*t^6 - 464451616236152639846880*t^7 + 46428990309509190682896*t^8, c6 = -38059341830230470571898103643651225 - 60802697439351169215335193049612020*t + 306907782422718633140882412470699604*t^2 + 350452937805251724377651403468219240*t^3 - 963553613419713777812910016932916800*t^4 - 1073907848531523127380729597230188320*t^5 + 2003891611171001180234808501351087840*t^6 + 1925073232043332626346743369726726096*t^7 - 4737928635455002698341677174065585720*t^8 + 3095917488055497753721277439726477600*t^9 - 978774537199645066029066153236296800*t^10 + 155737640585348222809204520763612480*t^11 - 9967046802486275083949405161786944*t^12), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:19:13 UTC · history