curve #4074

y² + xy = x³ − 637352421048928696212101333010045958480120232932914705x + 195682349641099974065169323688634390194939686417582443380627759988656001240951977

a-invariants: [1, 0, 0, -637352421048928696212101333010045958480120232932914705, 195682349641099974065169323688634390194939686417582443380627759988656001240951977]
c-invariants: [30592916210348577418180863984482206007045771180779905841, -169069550089910377592306295712869487443950755192062527057585693640767441843352366889]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 428177454037705020017159567275865585425893230857404351193992550565248483845677250700590
discriminant (Δ): 27898672880071397250402269997756377230491942813914356585194505563382055150318183713122713074381303582912360140115457802497387639455819721319288153821215910400000
Faltings height: 29.7073
naive height: 383.2811
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 167, 227, 229, 277, 367, 547, 617, 733, 883, 2269, 2371, 3943, 4973, 6317, 13309, 25301, 37747, 88873, 93889, 5302471
regulator: 23076576394292139916487.5937315055868522134506693065918998324606354639078552286
submitted by: wgxli
submitted at: 2026-10-09 04:47:08 UTC
last updated: 2026-10-09 04:47:08 UTC

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

(432018483169338435786264534, 982975130395123262337382272811428814023)
(438649327615708209141730274, 714194515428367447600140678341164873043)
(439755619745325120726590094, 667220722617621519734998777999437258003)
(440339276342141516043502484, 642066556680747523662008292187799447873)
(446865653753193506168465562, 324899632770018598361127901118291566635)
(448077203092294629962212218, 247273841778474990017935466392630879371)
(474583270019784545899902594, 309195404182038189176853793853341964403)
(475682658749564023478537634, 373456126003462480957166855773183138323)
(481593214310323563080195094, 659270439902494941689838691742978410503)
(192036116575911227230413642, 8964906511969640939697533882958129604443)
(230965794355935991803895194, 7797221139452358462390714007170947529003)
(245686559067371882871512094, 7343264280963085366440013442783486334003)
(312922467687827454059960874, 5184782793376131917245797184138048458843)
(345217836055165137568492674, 4098583166801419211311871471243432611083)
(428601930238183035939685194, 1116190459012160112444336191956315795803)
(541625670457616195467250304, 3060447586306082352833042700600682486413)
(61805152930715657105570382, 12511066056302660726291367789795821311263)
(693705253958426178265433322, 9347583135308879787739457192484198488283)
(2352214472391878705337766494, 108217792508707862364931548839528873150703)
(14813521189590297926328186351/4, 1764127517149321600817055800637496208855049/8)

Commentary

specialization at T = 62879/25160 of the rank-9 fibration of X808 given by: y^2 + (4224T^2 + 29864T - 18691)xy = x(x + (289T + 239)(45 - T)(5329T + 21775)(T + 6))(x + (169T + 874)(4489T + 5852)(7 - T)T).

last edited by wgxli at 2026-10-09 04:47:08 UTC · history