curve #625

y² + xy + y = x³ − x² − 47047239860512373923551296442x + 3933616682585190160331494669289172303089145

a-invariants: [1, -1, 1, -47047239860512373923551296442, 3933616682585190160331494669289172303089145]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 214830001287596349624206698595249377510923944541593010474955868696121214
discriminant (Δ): -19755076476754315868459409671256715103138590167284350219499075544428004109147912798208
Faltings height: 15.2612
naive height: 209.6794
primes of bad reduction: 2, 7, 13, 19, 622526231, 3675756619037, 1428933235076763360626659255797024274401157631
regulator: 1025696014974282830828.487196174684517823268436620553064745886950971432232592234064
submitted by: wgxli
submitted at: 2026-09-07 07:16:54 UTC
last updated: 2026-09-07 07:16:54 UTC

Witness: 22 independent points log in to add more points →

(-58024976337032999/361, -18586595227860985954662771/6859)
(791498052855361846056756505/14536673664601, -68631573561110382318037373729655198532459/55423961144350568099)
(6567652370902875549259111153/180923724722521, -3669448600661353035534858699411218006698989/2433566846336713519069)
(8792794536993445/9, -805704170973989778670255/27)
(152512418295356919/1225, 3325638764792625488814627/42875)
(147574541884448804017249/27112849, -56647058238306739874170297858231279/141176604743)
(7282888352047030279641/126585001, 1695503198805636195195926762552877/1424207846251)
(9754024832574742207600833/1480171729, 30446921323719165385255294434474098735/56946646929817)
(103223904672639224744691876094213/823754320152829129, -57033886503760212342233183856831175124003008431/747646936081370471740543867)
(-31400183607047733880344297327/1589113700688721, 139588929812717972604722628840372626218069823/63347934349894259751769)
(-28013896008270237317/385641, -632158785411890157834917253263/239483061)
(-206390422939139509545264633319/945853970958121, -56780693684719208902205443336167601839872851/29089492008930591285419)
(34503781509673332553490491089/290452157619649, 719255012628253081117471695216314619304231/4950076497221614367393)
(-186036427805921, 2499495965136489808135)
(-115864613745600415308403228232197291/605416447575227969209, 36268908016809048807530461167322692188362086643528723/14896399917975831272945639558573)
(-1424353141540465703/259081, 270003446520555880813142787875/131872229)
(67936255972355327002873/32410249, 17616046741198233218696132543970435/184511547557)
(9087582222171408943113/9504889, -845720022168645902384646937022811/29303572787)
(40477887504914950260467272941/112097167884025, -6923315716227737945979446772953118923938371/1186839414203063669875)
(56407368843232531184929364252113/182065837570693696, -339481746105843577301915468759027781612735038061/77685988299333315870496256)
(235371221685118861267593185/1273699587889, -1789660770180705090890775533845716793247/1437475701998531287)
(25718839984760908289185/12552849, 4102324577435129021229578912369233/44474744007)

Commentary

Specialization at T = 1/7 of the rank-16 fibration of X948 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 97707258T^2 - 9961276T + 6503447, p = 338797T + 722543, q = 1331(259171T^2 - 47235T - 2376), D = -12716840686473T^3 + 179128496547750T^2 - 5367669020897T - 5533394341440, E = 1124695(1356894976T^4 + 186730030T^3 - 206407447T^2 - 2883375T - 627264).

last edited by wgxli at 2026-09-17 17:35:32 UTC · history