curve #604

y² + xy = x³ − 70533871470555727826944636405210245041515x + 6949976567050306518996334891203075858929809744522422866258225

a-invariants: [1, 0, 0, -70533871470555727826944636405210245041515, 6949976567050306518996334891203075858929809744522422866258225]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 672916189200493189932374678182269172081492583885865619151065602052989614759908631144798052376110
discriminant (Δ): 1591567379854769932366973346865870374064343213115496228989012319224530864151866247958524555838312057089713009545950720000000
Faltings height: 22.4058
naive height: 293.7843
primes of bad reduction: 2, 3, 5, 7, 13, 17, 23, 31, 59, 67, 439, 11718406088935243351936391947315867132645945128746069272589709901238939675400392501
regulator: 5281924458658906845065944.56491797783361990269348272498513343047350030545076853063883
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:46:08 UTC
last updated: 2026-09-06 15:46:08 UTC

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

(191951832666557453630, 695293526215674942321450849185)
(519928877393321113130, -10527473257761266121966048097315)
(344666845741895840030, 4856343104686602457488221334785)
(-47488871441384501170, -3192562286817858584401760642815)
(167809959992829098320/9, 64128927157379014483561915277795/27)
(67095909872931884630, 1587292900458798780710558259185)
(126844652295859333430, 209779757816084643231976750385)
(8239793942914533230, 2523757404024566479148632053185)
(510292052043522648095/4, 1338482291331270497196882771205/8)
(105216107457788307302, 832745244614734177894275634001)
(199155468047990863640, -895462944634719753528188161825)
(12817691609760327430, -2459268390057147316197567653615)
(108542982816311687630, -756853689930854975441321636815)
(6816283346024501311130/841, 61597719848021324565264865670188465/24389)
(13614715038756219710, -2447897390894923203384232054495)
(-285329933777498574370, 1961067750510520510543030965185)
(-564067029396321907118530/2209, 299311447030323090539320751582398255/103823)
(-30433954740081577090, -3011380499191799406294854285215)
(2187625685050916666150, -101597155158291980353474565631535)
(2324369153721557050430, 111359120715353723932457949921185)
(-1767209835653227275163/16, 234217805433412659736800160123319/64)
(9014657829617409379382/121, -1932759799550740194458295271020509/1331)
(375158493370251116330, -5769736845946553121121005505315)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -215/89. 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 15:46:08 UTC · history