curve #436

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y² + xy = x³ − 85662285878324393186410215408681700x + 9305394083127186081227398401022012428368804281129232

a-invariants

[1, 0, 0, -85662285878324393186410215408681700, 9305394083127186081227398401022012428368804281129232]

rank (lower bound)

≥ 21

torsion subgroup

trivial

conductor (N)

6659238930698749498275990668738524591136206467171330889159250348722050046602270

naive height

252.9208

Faltings height

19.0000

discriminant (Δ)

2822864776306595502439005729534366424563694178962993624011022240824211533040745014935144313124315136000000

primes of bad reduction

2, 3, 5, 13, 31, 37, 59, 103079, 209563, 18283322059798488583734807181, 17266444769512957737328281593489

regulator

1433381768151416253076.2629708056391346889888746960591966158829737841285685991987

submitted by

Bhavik Mehta

submitted at

2026-08-30 04:29:52 UTC

later contributions

rank ≥ 20 → ≥ 21 · Jason Ma · 2026-09-03 16:32:54 UTC

Witness: 21 independent points log in to add more points →

(19660969398442454, -87342947765714641544049352)
(126230456546659724, 22440519097163676329067368)
(131002798560820698496/9, -1499110697604577069716267657524/27)
(98378199767480264, 42781110368327024090054108)
(559580290955541704, 369583414967705630468173148)
(-315097468016618296, 70798769197812717138253148)
(-327632986045763758, 46924989136515278595874598)
(-228449681979907046, 130200986440534461509512648)
(40767921013277924, -76686882970817902091584432)
(439744618535145434, 238058029208386291778819528)
(137114545559485454, -11732723292791646668908102)
(-260774407515264886, 117942392210068970890249928)
(285333492306937832, 89964208074208568844879068)
(194896930779094184, 3633328137141890322093788)
(291506410111396424, -95421871855010947969040932)
(-162442291570020536, 137601331628938628483225308)
(218546663542383752, 31977395547165784894000028)
(-3577960748916764/9, 2609302290046655971258552196/27)
(-8815399344098714/9, -2616256587078778253489116954/27)
(1825648433065214, 95650462360816657379674808)
(3595676155373384, -94854766526577549919192612)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 39. The 3 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 20.

last edited by Bhavik Mehta at 2026-08-30 04:29:52 UTC · history