curve #4004

y² + xy = x³ − 7122837686554889905496645x + 129363353005874415196596707510678901441

a-invariants: [1, 0, 0, -7122837686554889905496645, 129363353005874415196596707510678901441]
c-invariants: [341896208954634715463838961, -111769936997076007574172987241299766603465]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 52730095526316529723237209878616107447931486
discriminant (Δ): -7206338852272152955943343901355807894035042958187391355226664806656748760885248
Faltings height: 13.8107
naive height: 189.0345
primes of bad reduction: 2, 3, 7, 17, 23, 31, 37, 43, 53, 59, 73, 211, 601, 36697, 78193, 10736721060361
regulator: 214971018090307181044.003195070717263874792759139615255871228229541
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:43:57 UTC
last updated: 2026-10-05 04:43:57 UTC

Witness: 14 independent points log in to add more points →

(6229204281160469554/27889, 15546060781388961387032803553/4657463)
(-4435182696443642/3721, 2648586501949843243452439/226981)
(-236111804650709638988880674/43193668251721, 632172839240857704927913170872077079627/283876951353609987269)
(-144670048504430492569412/56206452241, 152308919334535422292726276887172081/13325369490844039)
(-19061254165186208807/15555136, 716113007904836147167884185351/61349456384)
(308236644486857332213/119333776, 14759992215023361711578853011977/1303602169024)
(1607917652714422096978/199628641, 68773849172143002168608851483127/2820553068689)
(16589962026551328408658666/265792833601, 67528388656231667405513072261825116033/137029761155829151)
(281546516145516190176056254/32459945153769, 4962776562444903162103338247249681058597/184936090501112811147)
(-2959007122999702344287318/691686632329, 5194693796422837991348499549809444819/575259863315485733)
(-34179131344983491726069412/8846692486921, 262094096036648495739188419994317213821/26313062484856120219)
(476787352276748997998086334806/54103815543948361, 344866511956805191107960699055944597906571847/12584670257079857987431109)
(81430378301370943562912394106/352386519785521, 23235550921920180559753534147841434756127023/6614986006339528576681)
(-1912361860674979242803534414/395705150890921, 56165402501908985260630330070608360257677/7871501005568352554219)

Commentary

Unlisted rational-isogeny neighbor of curve#3642 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 14: 14 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:43:57 UTC · history