curve #802

y² + xy = x³ − 29188703069051197770429450437433760294668752911731105x + 1914913739665654540832114459877752009552918529508658641086106271137300997150025

a-invariants: [1, 0, 0, -29188703069051197770429450437433760294668752911731105, 1914913739665654540832114459877752009552918529508658641086106271137300997150025]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 2707707219640613573728887421621760566601621577410675550720868877255002084906983107620003530 ☆ record for ℤ/2ℤ torsion, rank ≥ 20
discriminant (Δ): 7466481765281202467093889201536743587640377671795404644651773174185827020775677656599606300698189629156934350569799867861081758385394814800830326604032000000 ☆ record for ℤ/2ℤ torsion, rank ≥ 20
Faltings height: 28.9748 ☆ record for ℤ/2ℤ torsion, rank ≥ 20
naive height: 374.0305 ☆ record for ℤ/2ℤ torsion, rank ≥ 20
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 31, 59, 89, 137, 167, 181, 239, 311, 373, 379, 433, 809, 1021, 1123, 1367, 1543, 3359, 301643, 458483, 60388687, 332302241, 27570707443
regulator: 7383108267893498013215.45865347388241152092867314527000992229049438771244341178
submitted by: wgxli
submitted at: 2026-09-18 15:09:56 UTC
last updated: 2026-09-18 15:09:56 UTC

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

(102672928433494550029121610, 19350250119651018800400209587494837195)
(102728109853404198417727860, 22579174223061689145875422637976687195)
(87499856838290758648815660, 175570460465740496497781083075370589195)
(-99190226902144765491120630, -1958122780688980578626456400236039412405)
(69161475959805442059169710, 476445922241296707569339055479213208195)
(88986347136501450078286230, 148870052965353729538703195998905316695)
(21362140115975109689083050, 1140670415120704772214803792153664097995)
(102534744187930939119818970, 6605067719000827135665896993782367595)
(94514501111047889524776510, 21335680786690452120900001791572209695)
(5533675819443143997964440, 1324221419975655376763636997997279239315)
(8028994322511769832694570, 1296562916156846142784755679411303037835)
(91368246913747876997447910, 103679983389688649899061470508550294195)
(90361720579402087503911610, 123278114653328611041369760390327281195)
(94552266874320737911364010, 19114542739111332494917947635756397195)
(-128738826706683143982632214, -1881211391671814128876734951367136975733)
(-32162134778794982369150070, -1679409460456306338749719869013252346805)
(110822508602597072532632586, 203050026219210362179297574230466198187)
(462588438659914849416345780, 9348848131433773329439777008818219879025)
(756873227284794626441720490, 20332313060997905986897445332785234721995)
(76074715945544651618658630, 366965115500017307225816522845056546675)

Commentary

Specialization at T = 102337/261547 of the rank-9 fibration of X400 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 11T^2 + 1530T + 1112, p = -149T^2 + 74T, q = 704T + 968, D = (-13T - 11)(70T^3 - 65T^2 + 1938T - 1008), E = -363(356T^3 + 2272T^2 + 2597T + 399).

last edited by wgxli at 2026-09-18 16:48:01 UTC · history