curve #3124

y² + xy + y = x³ − x² − 595526764519545473538759784349237221738404650282x + 207404712817763552978197730613335214330524536966377034126822264871403881

a-invariants: [1, -1, 1, -595526764519545473538759784349237221738404650282, 207404712817763552978197730613335214330524536966377034126822264871403881]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 7882602549544199351275160289497567761080531375360072630
discriminant (Δ): -5066111468274005125739393545635032946710399655806922507181320810609494960253843205840030246924030839517365353390752509497488200105984000000000000
Faltings height: 26.4872
naive height: 341.9492
primes of bad reduction: 2, 3, 5, 7, 11, 23, 29, 47, 73, 113, 173, 229, 251, 751, 1861, 2351, 3919, 4007, 4639, 42569, 134999, 321569
regulator: 6131817773001.05627778498541356324890199962580209313293138217
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:23 UTC
last updated: 2026-09-26 20:38:23 UTC

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

(124724020384798220345041, 367516581253672140997126646108909079)
(15324897121776184029316609/49, 78023030459275713874593021865060671697/343)
(28612989853882985666849529/169, 733385789605154435150615291385098596963/2197)
(840079999716341125193632180425/395641, 726227720407682224612892525839435956289462763/248858189)
(-1584048856742125556157627017391/1803649, -557746031527477128709154863569311707980857047/2422300607)
(263385382871596577215041, -262341765294032260000723764046834121)
(27280404387326409806039629900561/159744321, 671707802680359923079182200461092289035487008201/2019008473119)
(466825184386555940312841, -176439935634864282815841693878298521)
(319445922309616026821601, -223078905661744788924733733911327961)
(678051177799205539879839686345375311005885921/1508904437607749867281, -10242265179139088732174481695367570720017471057776400624129966398559/58612818226198120621463239836679)
(25248366306029966967837632925/32041, -2735196341107935781267239092724000780378727/5735339)

Commentary

2-isogenous image of D-z2z2old1011121314-055, rational kernel x=7261995732809195702276475/4. 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 20:38:24 UTC · history