curve #3703

y² + xy = x³ − x² − 7611294031803572692929373436274099x + 251572227150820406941107992934837346654623559944793

a-invariants: [1, -1, 0, -7611294031803572692929373436274099, 251572227150820406941107992934837346654623559944793]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 38689313102431608020253080619974416595190
discriminant (Δ): 879231205253955348350554828866143838724692866639732547959045620293901871503060219460799303495117187500
Faltings height: 18.3567
naive height: 245.6584
primes of bad reduction: 2, 3, 5, 19, 23, 37, 113, 139, 211, 281, 313, 431, 547, 2029, 3919, 133319, 364943
regulator: 54208061944589.5760061468227627866176097049463495052177352814
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:46:20 UTC
last updated: 2026-09-30 01:46:20 UTC

Witness: 11 independent points log in to add more points →

(2364807755812394003/4, 3599008786208349425751663397/8)
(-693010585961443059943727/7112889, 156742416214813186243785276861903167/18970074963)
(-41499170854430919948/6241, 8566530759101273997186980175841/493039)
(17584056715193141387/2209, 1436699076112744762504647218773/103823)
(812788623068776491542/18769, 4582719319980408746759670624583/2571353)
(5145492978757711849337/5041, 367790134665995419637380971116974/357911)
(166610169206108633758497/5812921, 105776448848103657748393686284294054/14014952531)
(7631006540194929675791588/445336609, 105567426295238227787227957900493609843/9397938459727)
(15897188583598396723988687/218418841, 29436552935682957539932653090768275726/3228012051139)
(2713542141353455416534011557/15438311001, 126426800804935009012385897687464852076534/1918225580185251)
(11495174120583970392472298327/51651198361, 1149003424635537492826169665566348838906331/11738716200306109)

Commentary

Specialization at t = 5941/17480 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 01:46:20 UTC · history