curve #3136

y² + xy = x³ − 16807300719676114741159074415663219985838088x + 26485415853243610398964585891912523713720572243019985442333257792

a-invariants: [1, 0, 0, -16807300719676114741159074415663219985838088, 26485415853243610398964585891912523713720572243019985442333257792]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 16491334853285711166878253103438065925189351213350
discriminant (Δ): 822073273213488967035119739800754333198182281635839850752411690516194178448384474177543938046642789851281897504426422435840000000
Faltings height: 23.6350
naive height: 310.2048
primes of bad reduction: 2, 3, 5, 7, 11, 17, 29, 31, 41, 73, 79, 83, 241, 281, 5417, 488399, 41465951, 58252159873
regulator: 251537253420.431384735224928248833665684429377710258906982308
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:58 UTC
last updated: 2026-09-26 20:38:58 UTC

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

(103537673170890124687875300692/35366809, 10243145676995880519073995435048263319016856/210326413123)
(53669882994240811524368, -12398314172672731340761475016688648)
(2769370223832300118172, -34339734481312360650343220646136)
(5618288406133155590672, -330755727803203965189275759414536)
(406369413634260878673752381456/13432225, 256784252226667786908852069746410527872217976/49229104625)
(824222640856707743194832/361, 25065133723600799553259678479288296/6859)
(10368123201368124263470352/5041, 8893698072959163454389196802507871304/357911)
(118991967919531870238864/25, -29132349739590412368537301031996648/125)
(-1448461224197917863539989575054064/1165387179961, 268223674683005242216723605257284172887175947659880/1258071587770478291)
(1335045743559744192272, 80164927031290377516102577372664)
(2266893509302689197072, 5846479591535370715129069756664)

Commentary

2-isogenous image of D-z2z2old1011121314-045, rational kernel x=-19501586197533110816897/4. Source: https://dujella.github.io/tors/z2z2old1011121314.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 20:38:58 UTC · history