curve #605

y² + xy = x³ − 306326554639429186442670850869392366219940x + 65315896909451117664841196368012480933627909860539671599224592

a-invariants: [1, 0, 0, -306326554639429186442670850869392366219940, 65315896909451117664841196368012480933627909860539671599224592]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 57360850747856842585355023684652801319747635987610118789586775701209181928530683942995056902230
discriminant (Δ): -3339352782903064132366427465075405065893719131182818363958041004753758976658545135242078529006315627223597915294653235200000
Faltings height: 22.6192
naive height: 298.1918
primes of bad reduction: 2, 3, 5, 7, 11, 13, 331, 8293, 11839, 1575036567619, 40142416721029464614197091, 929632751394724171673762076244159198515917
regulator: 7693490628912856223051093.60736761365464723038917199988202753885446710003919820912188
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:46:20 UTC
last updated: 2026-09-06 15:46:20 UTC

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

(563506392554742649224, 8463712992951021095410063837788)
(400241601219406708974, -2612911179199270178237232380262)
(209127503722995319674, 3225001841654739830275636238088)
(1136742016166910499415304/3481, -67078074478154251940263366797748908/205379)
(452607953566435929474, 4403248044101934780196582732488)
(1310308726872051258564, -43744877783714877639028450677492)
(746868817299771028074, -15910477790784381686350974110712)
(284639825679838940424, 1088407178686795210587530946588)
(781352468906443004552, 17406656661216724600214504184860)
(568004805386836670064, 8635741131511158982878246299508)
(8254878397955450402376/49, -1474951283423688796631467280398716/343)
(1148553115271232002751/4, 8126623440475011339757181233809/8)
(25806691139485231197452136/80089, -5827523315176687897053704435805124044/22665187)
(1136287582122944540424, -34414501964324520791112664287012)
(3822692912396479787592, -233998424862140853482022274491492)
(685726884374190879432, 13330527788256035236647932410140)
(-438532298160574325496, 10738509478578312935619644062428)
(319838556093465666924, -243481888119344666275518431412)
(-9242492641667688261624/49, -3700349140005110778746755380065916/343)
(-201419699464008127176, -10901583525134410238755341222612)
(67311055597475653000392, -17462843693208210571335669419941092)
(-6408222565759469117730744/12769, 13888565554440243740952413044584598236/1442897)
(333954081899614258008, 511108977586980608666309735436)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -313/73. 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 15:46:20 UTC · history