curve #3239

y² + xy = x³ − 10363846234372609474021228275773377224259738x + 10412633142849975752506070356136680338076754827045078931231445092

a-invariants: [1, 0, 0, -10363846234372609474021228275773377224259738, 10412633142849975752506070356136680338076754827045078931231445092]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 235186508255669484903975893032399896944786213691049874204261617969438950
discriminant (Δ): 24404402540702114714152735230783157665020598266113141250940854330021962187436300392679766515254504089525626949949060592697344000000
Faltings height: 23.7354
naive height: 308.7543
primes of bad reduction: 2, 3, 5, 11, 13, 31, 37, 47, 71, 73, 79, 137, 149, 163, 307, 601, 1063, 1187, 1229, 3313, 5261, 7229, 7879, 15131, 59999, 363833
regulator: 51607178400795077394.065527005299917055763583465357002051022220996867655661
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:44:09 UTC
last updated: 2026-09-26 20:44:09 UTC

Witness: 18 independent points log in to add more points →

(-2607430220370914397748, -140387047515681414866483277225226)
(-1605621491785584326188, -151372793043869629484835629808106)
(-2819922555961640428588, -131201968173035899303243763491306)
(875074463835213331412, 44873030649530301538628591068694)
(-2467740801703347504172, -144775716117584701878214589749738)
(13315637656440444494452, 1494441905162102144306674307907574)
(-1762195702769915024758, -152327124138187809062915466865846)
(-3041170932937489746148, -117490087567011259916929625833426)
(193937709055224344613248/169, 12175020573097089290719700040789038/2197)
(1137909325716961985492, 9639692990949146169233800248854)
(3675715656247791166436, 148257394272207471703227166795334)
(52674857918538925070492, 12067240814424463070666174122025654)
(24178122670442307993524/25, 4496666863426713185668843068346558/125)
(3043510109618488310996, 84036038998028036136847099902230)
(1003524686553582969812, 31982322815500336822454512655894)
(673212038823402740957972/1681, 5481850155437583218016483153847046374/68921)
(644529864811619733766772/841, 1317924596196559752853144019460116206/24389)
(22583923428257353747412, 3360790983252352570792019048207894)

Commentary

Voznyy (2024). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 10). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:44:09 UTC · history