curve #611

y² + xy = x³ − 274386427944454351871655490254942322550862838x + 1722908657811417707084934239543814120584585527478509970094236694692

a-invariants: [1, 0, 0, -274386427944454351871655490254942322550862838, 1722908657811417707084934239543814120584585527478509970094236694692]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 3261524370448400850741938215757874717270759482136046406283953702184984006167703424017646575849989807834550
discriminant (Δ): 39755843575447047030216301023066961976032621353051073652934586644921344906295501718761565238591425945056760471223203103675480494080000
Faltings height: 24.4321
naive height: 318.5829
primes of bad reduction: 2, 3, 5, 7, 13, 17, 29, 31, 37, 149, 179, 197, 164839, 1928184310704543822344487145271052708611, 36146442251229899097692959698340776901842157
regulator: 62840125966300404664264429.4698538261943122848784383516596302332252694039048030969157
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:47:31 UTC
last updated: 2026-09-06 15:47:31 UTC

Witness: 23 independent points log in to add more points →

(13931327861232588316892, 777277231251059977851551668485674)
(8501288250596266104452, -68372600090559784513246093156006)
(-18151174719521189171668, 850391242221356282136869624876954)
(12506124531735490257932, 497388280134707802782990279049914)
(-16773672142692344404234, 1267281360669497239049439845792030)
(-8634152554693121282648, -1856970391226519473525942794820706)
(-7682287427591701874932, -1837779735720384631289767589088070)
(6175567730095669712972, -513749206209458734790692533924166)
(-18820724513811079668868, -469456258856685159018055371581686)
(7432835126926978080512, 306727407832473525708537659979674)
(52284876862151865019792, -11415264010899747881154216843426026)
(7632167148988685777492, 270775044871332276434122218046874)
(787107621085720472176748, -698161265826475706262830549467052230)
(303732816624282277895128/49, 174983437221505632642482213055287542/343)
(5731568474938048271456, 581834262901713503972308269838214)
(-14288792702804080042228, -1651127614927713893688392027546566)
(57451704129368436872366, 13251025747349250579853260556315604)
(2399572290067913365197902, 3716981813550266908815105591289470644)
(7901687288717658953948, -219426266594904001082487373519318)
(6918094594964402301807308/529, 7412800688603426771069140893488397718/12167)
(-369425548562352267238804/25, -199640569004651252745606359956381934/125)
(-3666429711921213854368, 1636960741753363915133688637974554)
(10543745235850982171372, 44745566241906523178529420871994)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -2446/403. The 6 points beyond the seventeen generic sections come from the surface's rational quadratic sections: writing x_tau = a/c^2 and y_tau = b/c^3 for a trace of height 10, the slope of the bisection is forced to be kappa/c with kappa = -b*a^(-1) mod c^2 of degree 5, so each section's point on this fibre is obtained by one polynomial evaluation and one square test rather than by any search. Claimed lower bound: rank >= 23.

last edited by Bhavik Mehta at 2026-09-06 15:47:31 UTC · history