curve #612

y² + xy = x³ − 4176582334201855248963071290816433543638146600x + 102681851603100829386828612635316022722738097004379031840980115560000

a-invariants: [1, 0, 0, -4176582334201855248963071290816433543638146600, 102681851603100829386828612635316022722738097004379031840980115560000]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 322602209191868708697404894366523935239392358711653542866101849627213446082633599031007395892472925210
discriminant (Δ): 107941509298485682587711104636524800379500922558245365430167053358366222380718250222801737930831670091202220934299800199778516881408000000
Faltings height: 25.1011
naive height: 326.7511
primes of bad reduction: 2, 3, 5, 11, 13, 17, 31, 41, 61, 67, 101, 193, 1213757269, 9023583685455464489147521, 3988608021162182130466712151822378082663888842281573
regulator: 123910445565995663220169923.36328632061038312084919721795546458613328107479163582293607
submitted by: Bhavik Mehta
submitted at: 2026-09-06 15:47:42 UTC
last updated: 2026-09-06 15:47:42 UTC

Witness: 24 independent points log in to add more points →

(20070421145620625197200, 5190461297268686244726659330525400)
(27215595749393481667440, -3028524081187111683231571919098440)
(44642834209092433538856, -2280301022590137675493216046962368)
(82108201477544107555215/4, 40475738456217692477736538410443445/8)
(32149452668028335141250, -1279172285334170220878979982529100)
(2555476058802396455197950, -4083853365295080501528284844888326850)
(43750464821209832290500, 1922845966667802311160457651170900)
(77763822804379215397200, 15752751382885317927941901225725400)
(-74278976257931604574650, -1757779908292902630594829598159100)
(125756668352345611450800, 39575974271233278460161120936835800)
(-38265534288453270101100, -14369086780176955049492154286242300)
(2001299630014701947406, -9712428868653440713188022868686908)
(40568942256755319395400, -110790286684906815294087008106600)
(1563633352351698024111600/49, -472595468281702955418628797262517400/343)
(29078147691850450276260, 2412731244896250092605721184784500)
(22990521812455112869200, -4337281252284613413416387679618600)
(128283778911566926909200, -40963707528208007919064772784930600)
(3337551821577675884693667600/5041, 191929042268803209835206363434009927447400/357911)
(22271804218931686598335080/361, -61285000940669770987730635576415193720/6859)
(-29134935233048747619600, 14129234915669435524215845126393400)
(15303370429641250549200, 6507688490231506191360859924061400)
(11614887038420094729360, -7465804236668209627958518428288360)
(33497986736473686770000, 602784555675426715826152865533400)
(3090588197649699928188486900/25921, -150452281364987937939619473174995707086300/4173281)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -3422/429. The 7 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 >= 24.

last edited by Bhavik Mehta at 2026-09-06 15:47:43 UTC · history