curve #3207

y² + xy = x³ − 194571228881510954052824732496450x + 211872838795725964920854163839521046715980990916

a-invariants: [1, 0, 0, -194571228881510954052824732496450, 211872838795725964920854163839521046715980990916]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 25369082817839617747015394004038159673868651735600634772174
discriminant (Δ): 452035982795747070511150179730472582897708589488717563990299065699756677159326623335737310839508992
Faltings height: 17.6206
naive height: 234.6587
primes of bad reduction: 2, 3, 11, 13, 37, 41, 47, 79, 83, 101, 373, 433, 467, 733, 911, 1289, 1447, 1949, 2011, 29629, 31751
regulator: 85805842332195324.813846195694815110583521957094484789307042872401395
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:41 UTC
last updated: 2026-09-26 20:42:41 UTC

Witness: 15 independent points log in to add more points →

(1715833272333801972/49, 2067584443541015776493800242/343)
(-1031700937567825836/169, -2378620581721364526927287178/2197)
(115018223996543973716/3721, 1105737108029117551254116038358/226981)
(3179983030768454184/25, -5636800471249830277418106558/125)
(-2181504880021970296/169, -1662502590368954947229469754/2197)
(-49221084312011976443414456373288364/4127848608236356081, -7670547989487975717615849341287163321651021367432878/8386594428876980664722775479)
(-40998541119059968169388/3052009, -3378570069922954332575650140627078/5331859723)
(44497527978517621957716/546121, 9249796507862907899593747385794842/403583419)
(77935390874667434881191794121604553585237455106/5828390615467915081650070369609, -447271397282264050538052415872007108822572545319125572080204984233372/14070935600280373192348131085328641339476910523)
(55948543399703789069213316/84732025, -418394925943226853198854430218221734954/779958290125)
(-38240381897144887089637394506848/5980720321403689, -505513032792128962177429667000739295440849917466/462519705800031893046613)
(437107598646611797882986852272525/400638356536320016, -7560617303188050636965009490595425111095192284369/253588052550675691182079936)
(3688971648692701958886/156025, -183120080409493765504194784399194/61629875)
(167792993899535990579355320966437446905656065366/7015886561567206891834162393225, -56482635005461170001094374353536438594106377683956269328094213547622456/18583342772479926359172729446179801865222070875)
(-69632802922580518213304286/5073285529, -196852012206461973428376604850373988122/361354908374083)

Commentary

2-isogenous image of D-z2old141516171819-043, rational kernel x=115715265331868383/4. Source: https://dujella.github.io/tors/z2old141516171819.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 20:42:41 UTC · history