curve #3167

y² + xy = x³ − 28869472695885471661757372297624036951x + 64622389720504351007385122791720566287900055948843115881

a-invariants: [1, 0, 0, -28869472695885471661757372297624036951, 64622389720504351007385122791720566287900055948843115881]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 18163244794012677234091814188102495252599967918193010
discriminant (Δ): -264140788437664607704569436217166839328753753603810074206585891444636212832398456372246807060542706712073070182400
Faltings height: 20.5041
naive height: 270.5395
primes of bad reduction: 2, 3, 5, 7, 17, 19, 23, 29, 31, 37, 43, 53, 79, 101, 181, 563, 601, 1237, 8195901717302849370763
regulator: 991282319595.69203155799351508799349546295519812069175046112203
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:30 UTC
last updated: 2026-09-26 20:40:30 UTC

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

(-1659604692679081274, -10390537676796644701248417083)
(-1914852673719792422, -10624597004918268380011532963)
(4109357091269126998, 3921909384565816504004501077)
(-3144281441798937770, -11149447531284652239735768491)
(5007892717428628720, 6755742471452922425467651459)
(1038004217878312726, 5981150039388949374785096917)
(1556385582455515126, 4843598619699546936644237557)
(4872117816146473858, 6294360179098678803863858545)
(2755114884223377228, 2448838642318405312754881935)
(3006346183195489426, 2236620292287813887263246957)
(52247998345704563941/16, 145531464673032264371820149623/64)
(1202117295854335722694/169, 32525186995722548118433094742649/2197)

Commentary

Eroshkin - Voznyy (2024). Source: https://dujella.github.io/tors/z3old891011121314.html (curve entry 39). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:30 UTC · history