curve #607

y² = x³ + x² − 830856671829334660247112778126062778423140x + 290909951187999109200265779335518757825576340394243607947394400

a-invariants: [0, 1, 0, -830856671829334660247112778126062778423140, 290909951187999109200265779335518757825576340394243607947394400]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 30850735333146160393655147247586815539031998716975145138749640447746073755426064682136055267761829160
discriminant (Δ): 148240876780830806794728997651430140140180140071341641121381593487446266320743344635443446881389656011769841682992221380000000
Faltings height: 22.8984
naive height: 301.1834
primes of bad reduction: 2, 3, 5, 7, 13, 17, 37, 47, 67, 103, 10141, 152151799, 1238346295113173, 1035343180402295367665723832883252361400522861121753731089879
regulator: 17672199467394920619640243.5627574016023183012566686329374987157758203187867877097486
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:46:44 UTC
last updated: 2026-09-06 15:46:44 UTC

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

(504656244572404018908, 371145125927674321124470621416)
(548618321122343062950, -459382723665638824449378104670)
(2153535977123291626770, -92136426394528380828543029090550)
(186030903017223152130, 11949183994941765503746397150490)
(-561688998977060049030, 24091143514991144910715723026750)
(1270544443189769540970, -35864859549228223769440161863250)
(79332039964713783270, 15016513524907494782669848020450)
(2998765899105493924205/4, -75632105514975639540182512510725/8)
(1360516349970382737855, 40973626582645710089745887888610)
(1771137644821035136095, 66145904472642630306112353980250)
(1732040641035850815030, -63623056773077661238922387439090)
(16950911503675301000021210/19321, -41367577756200176652266657468870655510/2685619)
(1199050607024022272970, 31915084534094569973137808267250)
(11088866342341872638530/9, -910669644657168629253997792560850/27)
(18835191593260255877736/25, -1202781715135435100480878338794784/125)
(-473203544661046560030, 24043996778287393742179556247750)
(4565736311935291581720, -302778036744886510482016079829000)
(-893749504337131779160, -17876553163036370281912515526080)
(288667423275522610470, 8667354270921298760628840054750)
(-1031402285398826763630, 7117674386056303993386153708150)
(2613123393960944685270, -126345822953294533048652201741550)
(1964521219587948433470, -78996435542742080277909875144250)
(546265184353792173720, -224714225882098796209587039000)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -1335/338. 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 15:46:44 UTC · history