curve #1927

y² + xy = x³ − 1035393398775308754670427378251x − 326949596064874283722508090895752774295421519

a-invariants: [1, 0, 0, -1035393398775308754670427378251, -326949596064874283722508090895752774295421519]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 30028829356595171319917435366264454669457035196770
discriminant (Δ): 24859798555573819912874887871678366552183915396453418330851394828457954697888784564131160000
Faltings height: 16.2529
naive height: 218.9506
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 109, 139, 229, 337, 373, 601, 1279, 2251, 2671, 2857, 6781, 15289, 146677
regulator: 6195514592256.214920766075012282534080847804528916234829444887898
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:16:35 UTC
last updated: 2026-09-25 19:16:35 UTC

Witness: 13 independent points log in to add more points →

(-1494959723630561/4, -22362386440211928980539/8)
(-235900658781491755352566/445505449, -80252254262482004708478021258819869/9403283512043)
(1843380435284780, -63469111459583693423791)
(196315756356652181/16, -86675315018928737462257537/64)
(8746899183529396706/5041, 19929554356036165519680007987/357911)
(-480400624438210, 7719125608406182934201)
(-63967273046799675792296785084/82601722925401, 2426885002075265453339411099526703693782383/750729806291930333149)
(1622410358882474, -47578892302856415340147)
(92477112989087639510843384120/53260789404001, -21676087513129361220487017233141697121824409/388697187809609893999)
(-1036508675669164255299090971591217095333340484/1421881062783081067608776365801, 10783156118017713199466391879565357925784129072442447486101135897167/1695488721642054229514120237427879872836163701)
(174993160882297690664342516477838099770672009092781610632394356786/21863106349491622054082031405713336780451274695721, -72585973312810890648572199826318709248688186485072026930812102791003613528409027459749141783930423/102227514327673429971749969985901000527303034094780020664770512337168544981)
(-376982951501591962494110882302/530033157957601, 86002456399142170780875142892517484967181973/12202661347387813184849)
(-103549726811848385005926692048425820318807995396/280792813162255321581643377563569, -10229742011200547151410140879826694859272057682466333244724174562733799/4705209680594895605903242244060163545233276747753)

Commentary

Rank exactly 13, torsion Z/2Z: 13 independent points; 2-isogeny Selmer bound 13, confirmed by a second 2-isogeny descent. Specialisation of Kihara's rank-8 family (S. Kihara, Proc. Japan Acad. 77A (2001) 11-12) with u = (s^2 - l)/(2s), (p,q,s) = (5,13,575/3).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:16:35 UTC · history