curve #405

y² + xy + y = x³ − x² − 16548893165735762025678539x + 26839119952931669744569431573523699419

a-invariants: [1, -1, 1, -16548893165735762025678539, 26839119952931669744569431573523699419]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 240304052236271865792163224043927979467874502821068908703531542
naive height: 185.8890
Faltings height: 13.4119
discriminant (Δ): -21126847178478909706119014513238013838343083475096497271069314604074275454976
primes of bad reduction: 2, 3, 13, 41, 43, 47, 11953, 12251, 74781881959, 1131749254105640328087296214832246279
regulator: 4640709170520558901.4172077288912774504840462322039265644301731391197884209
submitted by: Bhavik Mehta
submitted at: 2026-08-28 20:28:59 UTC
last updated: 2026-08-28 20:28:59 UTC

Witness: 18 independent points

(-3306824380777, 6738175697315251290)
(30189185258932871/12769, -1390580896453340819530722/1442897)
(4369528301117614763/6806881, 72095243229002689456392010794/17759152529)
(1047908035222739/484, -11460156637937551294845/10648)
(10912976686778979919/1347921, -32206576579933461969079244854/1564936281)
(792737529466027007/63001, 672921097931015116803137130/15813251)
(48157403470146683627/15124321, -149123370299841519846785247702/58818484369)
(3407404363506107/961, 106324670147184170633670/29791)
(5779643108996866883/1885129, -5741500542469560803558747406/2588282117)
(-2493593895740149/529, 3992508993373073837238/12167)
(423731527338700710563/20241001, -8569129758536490184358041766226/91064263499)
(106206734690790299/841, -34594380527848246207318926/24389)
(3842127816779, 4469163678560637066)
(2042955394907, 1247830848076410762)
(5499749719694097251/2468041, 3929861750163129309397079118/3877292411)
(36951547614887849/25, -7103089895787010745805378/125)
(7063245280319323154867/205664281, 589635975504701258328782671968930/2949431453821)
(7293197365467156371/546121, -18876049844717598811972197090/403583419)

Commentary

Rank-18 curve from Mestre's quartic construction, specialized at T = 2383/4. The sextuple is [0, 113, 550, 753, 868, 1058], the canonical primitive representative from u = -3, v = -1/2 in the Mestre-Fermigier family, whose generic member has rank 12. Rank is exactly 18.

last edited by Bhavik Mehta at 2026-08-28 20:28:59 UTC · history