curve #316

y² + xy = x³ − 1185784901231914789559558181x + 15709973088316374222740088167179617119361

a-invariants: [1, 0, 0, -1185784901231914789559558181, 15709973088316374222740088167179617119361]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 350481036484526026323524826173099607322997042184041024971581818910
naive height: 198.6342
Faltings height: 14.2954
discriminant (Δ): 89176446007892832942170673747379791745757428321235958497602360325335450136985600
primes of bad reduction: 2, 3, 5, 7, 13, 10667, 3889864799, 17205539371, 84895016131819, 2118237190297826516065951
regulator: 92302900417207424.6618806031761127297795276055382073375398310158206493746
submitted by: Jack Cheng
submitted at: 2026-08-24 07:32:42 UTC
last updated: 2026-08-24 07:32:42 UTC

Witness: 17 independent points

(4595707826225396946, -9852095564070173687393395353)
(27330565789098, 60964201325242834047)
(19528601652684, 895714120900637703)
(144365726817442/9, 771518233290848301245/27)
(20214959284656, 336313418519618847)
(239787319553506/9, -1486143331484671369859/27)
(-39296468065374, -40311743329885274793)
(3905575201413379/324, -328242264546675851387821/5832)
(18061975170018, -13594241571089606505)
(-76369704710221/4, 1417375645728035493331/8)
(12275980540434, 54802250006833328103)
(8816052662491586/529, -292243788677002214771279/12167)
(18370790904846, -11227234400360350653)
(349519770477602876034/11662225, -3353820769946805438405452648607/39826498375)
(46544298010424407728000/2381147209, -27245787719894335000213207371387/116192840357573)
(13182733475466182730/124609, -45591976984664375625274815753/43986977)
(3902300440344891/100, -169885612449259904662569/1000)

Commentary

A new canonical curve from the Mestre-Fermigier family on [348,-600,-216,492,876,-900] at the fresh shift t=2/9. Completing p6(x-t)p6(x+t) gives, after square normalization, Y^2=450555686361*x^4-8079008597280*x^3-158594291989460424*x^2+3160507498316956800*x+13015610394173754277264. Its 11 generic construction directions are enlarged by six independent points found in hyperellratpoints(Q,[1000000,2000]), with primitive-quartic sources (5618/9,140122690848), (9410/9,610042859808), (-4258/27,848258843264/9), (-33127/45,6138313886721/25), (7904/45,2379994816884/25), and (-4034/81,9044700913696/81); these map respectively to the final six displayed witnesses. The ICARM exact Cremona/Brumer verifier certifies all 17 points: F_2 rank 17, torsion rank 0, no halvings, using 18 good primes. No parity or BSD assumption is used.

last edited by Jack Cheng at 2026-08-24 07:32:42 UTC · history