curve #343

y² + xy + y = x³ − x² − 43285881523530475258933144164152x + 109614989666789360127642358536641482836869473979

a-invariants: [1, -1, 1, -43285881523530475258933144164152, 109614989666789360127642358536641482836869473979]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 50942599871541069354056504921648280000071055124980383441294377459196586533970
naive height: 230.1498
Faltings height: 16.7919
discriminant (Δ): -58182934520641038384251737838573035416513937358706673216587586437986300447298128185328640000
primes of bad reduction: 2, 3, 5, 11, 19, 8768717, 151942966871, 459535661555689, 645058738193217419, 6857363496744081905501
regulator: 142470474246966436202.2068711643981879579231795504385081941117192671659308500
submitted by: 7fff-zip
submitted at: 2026-08-24 19:19:13 UTC
last updated: 2026-08-24 19:19:13 UTC

Witness: 19 independent points

(10300201881431681541/25, -33053204234615102579744642779/125)
(67440588834602259, -17433470796488479171879727)
(26305577587794169, -4144149891548775086682543)
(1253025071323627581/1849, 22571077665952629155694830611/79507)
(2898164418726281709/2401, 28607049444422613519402045433/117649)
(2162911440499081835413/683929, 37317821686993197983715011926923/565609283)
(805006435096777789088877/214300321, 14273685517551851498901877985847239/3137142399119)
(218524850133302208941613/57866449, 1095031012143441974860533435329183/440190077543)
(103436647643503776191773/27279729, 152189280606669150321755032574927/142482024567)
(3797385773917737, 792825778477831830181)
(4671288627586987, 96674919376191279478081)
(1638640481144528877194997/344436481, 681370476719222173250554948242143599/6392396650879)
(88367861778264817, 26198062363001806352982081)
(32713710531533653/9, 468748128376139293239187/27)
(1098786131269157487711389437/20335045201, 36164246426922687171551889485693092857281/2899797780707801)
(5494276783500143547247/1418481, 13607441002995250935553485117151/1689410871)
(817560845382286558572957/216119401, 5841488894703062831449953932135461/3177171314101)
(49601422640478456723933/29997529, 33895265503831280222476187784628373/164296466333)
(17248096730967104320958900253/6647182273369, 2082624846828637460677989855139852181529757/17137851750569509597)

Commentary

New in this search: Mestre quartic family with roots [0,115,206,234,731,784], T=14639/10. 19 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [10000000,10000].

last edited by 7fff-zip at 2026-08-24 19:19:13 UTC · history