curve #406

y² + xy + y = x³ − x² − 21097251078837597697404902x + 37018745916232850579640184758127094901

a-invariants: [1, -1, 1, -21097251078837597697404902, 37018745916232850579640184758127094901]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 2644553355450364179430274416851803216811334386659749432662237930
naive height: 186.5472
Faltings height: 13.3984
discriminant (Δ): 8969214692406883015730398815550145450361500168920974846388502144587468800000
primes of bad reduction: 2, 3, 5, 29, 47, 3264101, 982147597, 1233077721715118924761, 5453598607079233734187
regulator: 9383778911288440110.1357955156395791959854566005916675790171404039597616127
submitted by: Bhavik Mehta
submitted at: 2026-08-28 20:29:07 UTC
last updated: 2026-08-28 20:29:07 UTC

Witness: 18 independent points

(774954654401, 4597252669544889099)
(12998944309897769/12769, -5878204090838967977438997/1442897)
(173463963061750709/61009, -2078635754840004602860623/15069223)
(-39670626660931/121, -8818854028811383937979/1331)
(3528660775209957061/938961, -2991316092457691829405345509/909853209)
(937634068670048801/63001, 868947381915949154789355249/15813251)
(238100297958593141/96721, 1124136951756921055763889/30080231)
(8629332410790761/961, 712256125210844543384709/29791)
(4550481028769441/1849, -4611156367817102527647/79507)
(-2669533887520471/529, -47079725558352391977267/12167)
(2644115007895631489/466489, -3178782620894868414475633887/318611987)
(1882793609828342609/635209, 367307169483188807749466727/506261573)
(32009748639641, -179331156600287048181)
(1972047015990865511/383161, 1908806134510994025910543041/237176659)
(14666508037349599411/86436, -56147778119232515764106975909/25412184)
(1982097829371409/9, 88225452343984517038073/27)
(14529639536122041/5041, -141029179945811070820211/357911)
(84290081689321481/25921, 6983190880364833683634779/4173281)

Commentary

Rank-18 curve from Mestre's quartic construction, specialized at T = 2357/2. 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.

last edited by Bhavik Mehta at 2026-08-28 20:29:07 UTC · history