curve #3219

y² + xy = x³ − 13397943136358263756130318190185x + 18870818293000775719167980823166925265792205401

a-invariants: [1, 0, 0, -13397943136358263756130318190185, 18870818293000775719167980823166925265792205401]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 23989560829689213420672070983398343053293012701577866
discriminant (Δ): 81193164039726443329464018245958898147162114558075629677434784013166505005608479221831145472
Faltings height: 16.6129
naive height: 226.6316
primes of bad reduction: 2, 3, 7, 71, 79, 89, 103, 257, 821, 1327, 1579, 2543, 2719, 4831, 8369, 15511, 395849
regulator: 23049205338222122.729853443485517726608197094572861292143702920494237
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:14 UTC
last updated: 2026-09-26 20:43:14 UTC

Witness: 15 independent points log in to add more points →

(1029022500230176, 78572591474967918603931)
(1578163880681536, 40709276641363021726171)
(-4141305136555478, -57712347640134375649421)
(2772428055260890, 55098615113008600208131)
(2082907841381770, 918540150707067712339)
(2054126279090674, 4123475144428102233115)
(115634161138496260/49, 6821596742053021577949397/343)
(-26892431106076454/9, -4846927644106656067145023/27)
(3487894012418428, 120713935554030864221923)
(784680557315050, 94025842467281956796899)
(2074435621130440, 2128765595701496186731)
(2085104685677122, 183713608378515394219)
(344467348644442930/169, 12089068728914504633365327/2197)
(709694235667821055/196, 366416357361358895030672471/2744)
(962081214366697076323522/421768369, 115631260259264430062170152953273099/8661856994153)

Commentary

Dujella (2009). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 48). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:43:14 UTC · history