curve #3615

y² + xy + y = x³ − 7146369925765405477086551x + 7291916366253844925956185473652201998

a-invariants: [1, 0, 1, -7146369925765405477086551, 7291916366253844925956185473652201998]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 1307294923137952127239328526450558626015820657750718325629240650
discriminant (Δ): 387719886337471451977789562313536262498893766712219458873300045719822437500
Faltings height: 13.1323
naive height: 183.2995
primes of bad reduction: 2, 3, 5, 29, 43, 34549, 4387647745637257, 46105063745128339757646112073002407301
regulator: 281234330321755701591681202.444225372625472990565866980177968634053404326660892
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:19:45 UTC
last updated: 2026-09-29 19:19:45 UTC

Witness: 20 independent points log in to add more points →

(2041413485312, 1100246106266938581)
(-65586127374588718911432/73249798609, 71404618502477435978460686288138003/19824838244130023)
(-16298101702046029029786/226335208009, 300847531230034540963256759555721991/107678296204657723)
(-14455530512746682852298/8739819169, 3120606762147355128190442612063836/817059474652303)
(-625180040188830786954/2011612201, 277834837112146508136542324489911/90222818827051)
(168077729877291891/11236, 67874275203417993108397891/1191016)
(3871102835777394813/1803649, 3286266454754986324267613642/2422300607)
(30028053341363809407/15665764, 49460088525381712725645658099/62005093912)
(36005790246380903837/20421361, 38370683683884866294779450644/92284130359)
(38232652143580369677/21613201, 43301150100203113370281116614/100479771449)
(49553431457339747628/11580409, 292428337639577422494496582282/39408131827)
(60216414583507701153/71351809, 822406471285930267473750623468/602708730623)
(95610520424204372349/47018449, 348492574010568966488885134508/322405504793)
(192070466073478011672/199346161, 3210121541961934883026909787299/2814568447159)
(273632795887019843767/191850201, 66872811359761797366019170566/2657317134051)
(7390766526912451926879/14922769, 635371750435323153302869397680274/57646656647)
(88566359032326448764498/39293943529, 12643896994352385963995301895398248/7789120543923083)
(11249564565565406478988557/6282898178041, 7658718573766409199671360873687778596/15748530369030407411)
(1067171200238375898959130777/755152653361681, 2544290409544977539321565969365665789586/20751625875637781708921)
(9911259344904226496121728637/5955480387154081, 41263657900978846025732319539002278995594/459594897166681434798479)

Commentary

Specialization at T = 14/3 of the rank-17 fibration of X1008, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #751 (same T), 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 19:19:45 UTC · history