curve #3223

y² + xy = x³ − 14654189369839995769496109219560391538x + 19008878679275452006123327230168503570949134500981218692

a-invariants: [1, 0, 0, -14654189369839995769496109219560391538, 19008878679275452006123327230168503570949134500981218692]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 1518905128450311243438362902929463593569897853087633327924112350
discriminant (Δ): 45304952459165919255314382335478619447459560354533223479690941713480522086892861457677617226875150545453056000000
Faltings height: 20.3451
naive height: 268.3470
primes of bad reduction: 2, 3, 5, 7, 13, 53, 127, 137, 139, 193, 197, 349, 503, 1901, 3853, 4703, 13633, 34693, 259717, 491423
regulator: 6974572572755664.499920951057816769606883148859314691362567056748818924
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:25 UTC
last updated: 2026-09-26 20:43:25 UTC

Witness: 16 independent points log in to add more points →

(7015812266231410952, 16171811732325153037621844474)
(14202884310794403716, 51729201166903825510053456830)
(221106882359692226276/25, 3013545121128313237963976102086/125)
(-434063802263819801692/4489, 1359267799692517323052016067033482/300763)
(14968584146028372685508/5329, 36172026858038356360264840374158/389017)
(2437398173762165019908/529, 85434020326881722601913520062258/12167)
(37203319685901343692985226684/9586955569, -4258433397283870514845653854648072333207090/938687580627497)
(885540984542806805856343100936509402177988/270794398061696264662729, -346843113118600010903691434706419241447686012503767332519866262/140915742636227615860196473287763067)
(833000864701173232149412/292681, 94456467390189919734712764944416054/158340421)
(2433688872161372860420327268/1628687449, 43998230115629446979209903415390780368262/65728939379293)
(-14413561802736011430288081628/27175852201, 23119399194928291062145500522223526505419114/4479966411187051)
(-2653276064290069948, 6261929395060343278047486974)
(-544142855172799897337508832586547187/194753234537868304, 530793161032312869765101345570159634542925926814300113/85946267152210112253462592)
(64302877790389736062226468/46199209, 359229975375128000428252646351787742102/314016023573)
(196316187418090315535341310482/13530575041, -84180982943941897394578146396621435502671886/1573890019344161)
(-6258434090598455032879975294449052/2141316115763449, 601688847960410735735577433060955309110099243478182/99088026390691166577293)

Commentary

2-isogenous image of D-z2old141516171819-027, rational kernel x=-12420185912395920417/4. Source: https://dujella.github.io/tors/z2old141516171819.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:43:25 UTC · history