curve #2970

y² + xy + y = x³ − x² − 16104043263800550674289475060800531785x + 24872405319381098775182406900478881695149129457377299401

a-invariants: [1, -1, 1, -16104043263800550674289475060800531785, 24872405319381098775182406900478881695149129457377299401]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 144739364313004453827525038927796126
discriminant (Δ): 40272953261685938927578784702341463563085268547081462240476624679162113271561435533081989288994877158413303808
Faltings height: 20.0727
naive height: 268.6300
primes of bad reduction: 2, 3, 7, 13, 17, 19, 29, 31, 41, 89, 109, 139, 173, 313, 2281, 2339, 2903, 6563
regulator: 102478392.624714793997852380575499306034321547221208859
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:31:01 UTC
last updated: 2026-09-26 20:31:01 UTC

Witness: 8 independent points log in to add more points →

(136350592450299414416375671/51624225, 324065411717538709957993607248065835564/370920056625)
(5167736285524917793558591/1885129, 2981156009749187681042798335550933996/2588282117)
(66558368728789616817691/18225, 9455147933400573233187659996308456/2460375)
(2140625267896547607, 456749749203100883185061500)
(811665033145819500001543/199809, -459662201547522492075947638446330940/89314623)
(2156229249005719109924583/954529, 136023912863404745191786643031598460/932574833)
(608301332889029030157136296612369999/445217216229241, -474434520081036295782641340252997690475542913662783628/9394159394580960300211)
(62526770471880377755171282348047/11655798855481, -383459257501677936301500052001072898122069522572/39793584984744607379)

Commentary

2-isogenous image of D-z6old678-095, rational kernel x=-18666384651289037877/4. Source: https://dujella.github.io/tors/z6old678.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 20:31:01 UTC · history