curve #3705

y² + xy = x³ − 74475283172642217970967566182277080104764970x + 247347677910499703177133729456351453272485365188696377089984756900

a-invariants: [1, 0, 0, -74475283172642217970967566182277080104764970, 247347677910499703177133729456351453272485365188696377089984756900]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/4ℤ
conductor (N): 15662834656502411939691591195371771384959399565490
discriminant (Δ): 7123832361906247191371462610390964156376565481484328767269317132582264854770161061912366467847004494651785981833610000000000000000
Faltings height: 23.9276
naive height: 314.6707
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 29, 37, 43, 53, 59, 83, 107, 109, 127, 157, 269, 347, 607, 3181, 1307863, 23637023
regulator: 23535455614929.693934462849057192885911045488528206208036582120
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:46:42 UTC
last updated: 2026-09-30 01:46:42 UTC

Witness: 12 independent points log in to add more points →

(-9662873523069354350924, 254478979291982856719720850726022)
(-5306102752051519906124, 702231665193734787878365835306182)
(-4500003457382048915291454930860/483956001, 3924978233451101789147565951534611492168382170/10646548065999)
(-20722315561156134184974260/17161, 1302181647400992693483240433126869501530/2248091)
(1883051960445228836428060/169, 1966104478095500013521664633234158710/2197)
(28295446937927571442055860/5329, 15571803272302098552738507543300041110/389017)
(74297587920633603034388740/361, 639861694273216760305089614685182931970/6859)
(317364771080981374324062307/64516, 84094696430311301507549844855870297929/16387064)
(1276529578273313214373850365/309136, 17380247482794805895225233588331109520405/171879616)
(1934778697896993418050603398740/369831361, 214448296867995806651366992897018674198725530/7112226903391)
(24951713764973725380369654005140/5059334641, 796740332664876848228377229270651371260550070/359865413679689)
(74014217019024291840717546212054172531000750256456139037940/10591590664163077422384337039058024241, -8998902468908999721087975365309704457055047897471497250108461482321300429462183114631370/34470039589181745927702152856328997986725083560565873961)

Commentary

Specialization at t = -30706/18375 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:46:42 UTC · history