curve #317

y² + xy = x³ − 4315862856022573416006x + 107904629186874422701928761288036

a-invariants: [1, 0, 0, -4315862856022573416006, 107904629186874422701928761288036]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 382724524351827852913185875134830026418690856518876504970
naive height: 161.0634
Faltings height: 11.2922
discriminant (Δ): 115017721248537024791512873609799594141037969992867713718881894400
primes of bad reduction: 2, 5, 11, 23, 29, 9377404282283, 556269352188281130391524829339614864907
regulator: 1522807205165362.956219376809242957391278748925554379651079502171632000
submitted by: Jack Cheng
submitted at: 2026-08-24 08:01:40 UTC
last updated: 2026-08-24 08:01:40 UTC

Witness: 16 independent points

(350823376408844/5041, -4304074360072412521722/357911)
(41613352696, -606442401259298)
(43754416876, 1682882181574762)
(-514519682456/9, -349744460202098386/27)
(42850357588, 1283793738135010)
(5575404693310/9, 13093570471884083482/27)
(24735711016, 4035253563971182)
(70005766238893/1296, 265514088956225527675/46656)
(-75671565400, -1087750532760594)
(169221880657869/16, -2201286064287084439973/64)
(431937596876, -280767657962153238)
(2283920593876/81, 2141528641062149458/729)
(32362547578701884/351649, 4616814020286500130366474/208527857)
(59381760629636064/1297321, -3695917007568875981110418/1477648619)
(5970160613604639/114244, 193370327124387817329229/38614472)
(-57052998624, -12977933128655338)

Commentary

A new canonical curve from the Mestre-Fermigier family on [348,-600,-216,492,876,-900] at the fresh integer shift t=781. Completing p6(x-t)p6(x+t) and removing square content gives Y^2=4001161*x^4-60808320*x^3-1036031133986*x^2-38029624675200*x+46823070345440729. Its 11 generic construction directions are enlarged by five independent points found in hyperellratpoints(Q,[1000000,2000]), with primitive-quartic sources (-293,6467696), (536,222888923), (-1169/3,119790416/9), (4739/3,42286639784/9), and (1631/9,8043119288/81); these map respectively to the final five displayed witnesses. The ICARM exact Cremona/Brumer verifier certifies all 16 points. No parity or BSD assumption is used.

last edited by Jack Cheng at 2026-08-24 08:01:40 UTC · history