curve #3139

y² + xy = x³ − 1234140270416378250678088578402933925437564x + 527447606280303914796391760482999986982123024359861103558856208

a-invariants: [1, 0, 0, -1234140270416378250678088578402933925437564, 527447606280303914796391760482999986982123024359861103558856208]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 5300423313342790652288129860235660003742350867406
discriminant (Δ): 119371036414019011111138119587532608188007883235958316759444193334079961840375049684246931168480927535384463941063441322606592
Faltings height: 22.9460
naive height: 302.3704
primes of bad reduction: 2, 3, 7, 13, 29, 31, 41, 59, 61, 79, 163, 601, 607, 1231, 1327, 2652103, 4368649, 823091551
regulator: 293669610165.376114848156883120742920880381402912823066066128
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:39:05 UTC
last updated: 2026-09-26 20:39:05 UTC

Witness: 11 independent points log in to add more points →

(709854151409081257848, 3013053332382683287650073999428)
(22744758292531540126629594936/33953929, -227497543993207940466707988963366484305876/197849544283)
(586908362296483747834588152/190969, 13394964856000011690760045225905544125492/83453453)
(4990896667677781057426483602744/7130958025, -1538061034694278823778016998444860244259466412/602173750421125)
(1062772792746641133432, -20401566229736295136350415456188)
(2776691497482834086472, -136047623182217818541217481035324)
(-40523127142036214257585608/32041, 46354443978431482597451747683995361644/5735339)
(1211429423820876546285345912/1216609, -22687719208292944209843735965200336499844/1341919727)
(1305666854674964352888, -33792424270074872172362806691772)
(7864353463742404222415928/29929, 77024420688208441252644606203221130004/5177717)
(82960197411147952439693459832/125059489, -1151620588266421982802376953254272996424964/1398540265487)

Commentary

2-isogenous image of D-z2z2old1011121314-050, rational kernel x=-5224161910020248206273/4. Source: https://dujella.github.io/tors/z2z2old1011121314.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 20:39:05 UTC · history