curve #733

y² + xy + y = x³ + x² − 4092116727688420072390649475x + 110721920693626166311255762161035719685985

a-invariants: [1, 1, 1, -4092116727688420072390649475, 110721920693626166311255762161035719685985]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 1863758222087171211274812966805615163460265969143252914506159374764670
naive height: 202.5388
Faltings height: 14.8456
discriminant (Δ): -910487017029411054460720427405483502710087627078532886223610114324812102227046400000
primes of bad reduction: 2, 3, 5, 7, 19, 29, 31, 3906657554604290739885947623090224418297883612623798396578899
regulator: 14034576273191480333.0919549616405694755926702995816898962686930855754613055748
submitted by: Valery Asiryan
submitted at: 2026-09-16 17:18:16 UTC
last updated: 2026-09-16 17:18:16 UTC

Witness: 20 independent points log in to add more points →

(-284059328390393/4, -1662528880525264969961/8)
(-18880592383436243/289, -1547588454223506075346196/4913)
(-59966681299287, -374796211356730610242)
(-59899555795157, -375394265881813959802)
(-54096058422278643/21904, -1126787910369638595643181589/3241792)
(13309824488003, -242104157129123841762)
(29337736818403, -126172711317081650242)
(36983402524963, -99831803598946430242)
(3341271432900595/81, -80229194980434290379410/729)
(47731852346233, -155391489063345215842)
(50112183248133, -177483209887680289642)
(317638540462123, -5555070620003287164082)
(945068986338883, -28988564551859637244642)
(1232240233907843, -43198629422350091235042)
(5146515210275827/4, -368759968428337122601811/8)
(2917677888666883, -157562397722586680164642)
(5608933607874883, -420041636038373995084642)
(29783439862833, -123548007763095492192)
(30700339502083, -118439665864725801442)
(34060610434423, -104194541351936854042)

Commentary

Specialization T=-7/12 of the family identified by wgxli in the commentary to curve #302. Starting from 17 reconstructed generic sections, minimized quartic searches produced a set of 20 points certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 17:18:16 UTC · history