curve #2742

y² + xy = x³ − 1200762149057538359683148051953717726878x + 15962988101305967882588329151972503212116946679909605925572

a-invariants: [1, 0, 0, -1200762149057538359683148051953717726878, 15962988101305967882588329151972503212116946679909605925572]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 270662653535270356603978350
discriminant (Δ): 721913531590592966763893000216812275944212979119045583069110613121160468844364945930425014999301325028865873187456000
Faltings height: 21.2823
naive height: 281.5649
primes of bad reduction: 2, 3, 5, 7, 11, 19, 29, 31, 47, 61, 71, 157, 241, 307, 580219
regulator: 6373674.334975949761686474489569252093857518357
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:20:17 UTC
last updated: 2026-09-26 20:20:17 UTC

Witness: 4 independent points log in to add more points →

(-4114999072350270948636522131207988/119401361430649, -167144582648038980364604225786336644517185663140630/1304709780679314673357)
(944955350155606487363352105480043858550976/411161442969917505121, 918476074967289465049234859168463344862577035308735671821665106/8337168372095986631579394759919)
(1913528408618805072532913250731424/66449642272225, 39302721571038954589953282058212877149036468946638/541675223173017004625)
(1424464915402967511202906955000884541519645910943518/77765172736848738105605164000129, -7325524991928213910643955858455377582597993712063568528567232812201942776148/685768782829219185855215745505745540446464781633)

Commentary

2-isogenous image of D-z10-008, rational kernel x=319985493971327392543/4. Source: https://dujella.github.io/tors/z10.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:20:17 UTC · history