curve #4073

y² + xy + y = x³ − x² − 856525067016695179253084598784877638459097790317028155x + 319473846427730289150973087825435625838292128304266567579822362149784628206258347

a-invariants: [1, -1, 1, -856525067016695179253084598784877638459097790317028155, 319473846427730289150973087825435625838292128304266567579822362149784628206258347]
c-invariants: [41113203216801368604148060741674126646036693935217351425, -276025403313558969826440747696166966248678240136220041051432950990248796061729130625]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 6298072849453555625678659198919335015893455426961399018547798525692978105732842174850
discriminant (Δ): -3875325092759126902665564147900677323007660241475496180140852653332021997497273298240149593392428486351535562900614983687751315080518572116501975499502731096000000
Faltings height: 29.9553
naive height: 384.2598
primes of bad reduction: 2, 3, 5, 13, 17, 19, 23, 29, 41, 43, 67, 73, 83, 89, 109, 139, 173, 181, 919, 1217, 1931, 2459, 2719, 3533, 4259, 4993, 8081, 9629, 100547, 234673, 761667509
regulator: 105835422220116849537186.071393799497699390644470968305137179535028445852126262
submitted by: wgxli
submitted at: 2026-10-09 04:25:39 UTC
last updated: 2026-10-09 04:25:39 UTC

Witness: 20 independent points log in to add more points →

(348236977567417424970218395, 7964326293900909109495251973039789764026)
(359869119140215442010412859, 7605396574707506627806400317532902676170)
(383042921914713232568961725, 6898460676090748416284919096821335817558)
(454890873648324593297936609, 4896634149682079246662534957269084084920)
(465477825442761513118232719, 4651378056682753646767694277689198393990)
(466148981958044250494374523, 4636522665466229293249370729478324858746)
(500357412620893514071034719, 4021620466592663658608763861436811133290)
(532387761697102999823141543, 3790597418931794428840253110282703216746)
(578172459486175670449305479, 4186656573244981412115948566573476367210)
(643287866215060914746894359, 5889531779124119033297467378276112457470)
(662573733086562023533142779, 6544869934567681159454795846759139625610)
(158165268526310124276254473, 13709778679657462911391210817430410620496)
(785809559062426833398207419, 11473580162854961283195366377062730122490)
(891519910611851702789571319, 16261968834824050858865984805866762466590)
(4731273560812127117148967469, 319650338974316931245870497112583907564540)
(1241682311156989382789440015/4, 73104093406876092476950118874845683126543/8)
(18178242024494531305925022691/49, 2491765040485197494274662091735900436151510/343)
(520175848017898393698147310891/729, 166557033546542318450407363974043844795866630/19683)
(1788121528293845087105493674647/2209, 1299000158801128923090360381980826453998582710/103823)
(19257724578244132107898325085271/7569, 79595680363386450357246417160025246663918179010/658503)

Commentary

2-isogenous to curve #4072

last edited by wgxli at 2026-10-09 04:25:39 UTC · history