curve #752

y² + xy + y = x³ − 367647057132661536131605961085494446124192297751927792319003x + 98299706024123968245526077311124299338038232423912535744363101436934755800391181044000522

a-invariants: [1, 0, 1, -367647057132661536131605961085494446124192297751927792319003, 98299706024123968245526077311124299338038232423912535744363101436934755800391181044000522]
rank (lower bound): ≥ 31
torsion subgroup: trivial
conductor (N): 118086809041464888793311247202094884684553044573177346565213826342066349036594569288948221822643995232870818552609029732700183942908086257866846
discriminant (Δ): -994005677097790618073395582743699276047907143367670871565278009974694508797586936378616465191972333501927687585778616436456160268240107728024609264941335753223652448378716931562388
Faltings height: 33.2634
naive height: 423.3490
primes of bad reduction: 2, 3, 7, 13, 17, 23, 31, 37, 41, 43, 47, 73, 89, 311, 9241, 938491, 163020307, 19965160070705630767875354870780624150699682679, 310159908337954032681514410956987460008335016509084620167
regulator: 12169008334690293321262617234857761763.55684995613335402579554227454070608121846873602805821336722897
submitted by: wgxli
submitted at: 2026-09-17 23:58:09 UTC
last updated: 2026-09-17 23:58:09 UTC

Witness: 31 independent points log in to add more points →

(71105554675963492526000845558, 269290675255312661174603378631020782861321040)
(-29704042557861724610106945056, 330445307249640665572931018435703810679069958)
(175101426740732876260582241005, 198224319312459426980672090698178741258477618)
(248378798255456892042084921013, 149355158680850764630077956833029679160249216)
(313900322947403427672774580018, 117579105889397601198987345355815654946667156)
(333644300759445321399900785433, 113036057509898777135980865422754832845580526)
(337344195504174583822768511518, 112544426509569291513494634176313983515956200)
(391217840641677214394952534118, 119775022330853746321023953719517878370351456)
(401963771980612812461862639853, 124362992830515375551949254364908031216155186)
(464097733011782059104565490608, 166240855799271737505324188103123394017945590)
(475134242728882677950838093241, 175729241082718683415630197857841349279480910)
(529405211752483370638248246438, 228126658076586731082043695240804220559709241)
(579189497967497021932404809974, 282236736482276291601963258814603820401596857)
(625994455340224943038994586453, 336842016882366329700990861656802279403168810)
(749358867473339221965452104495, 493552567740834272838256207863808118503287806)
(797654601729686318207740399225, 559065616589325930326866108761173159611508216)
(837492018316060116985328636506, 614661833404508719417509841535129164548977708)
(-107722374782128106191710128387, 369666738848101167311314042281532540472062370)
(-233000608579077662642766453614, 413898849914606078624628944133861443516843060)
(-660994466322383342336243257631, 229161184819056810524321051792070900954383558)
(-688879569869957765431042713827, 157012629314028111691246563988770194336931680)
(12634572579830709787101640264696, 44859120442512732892230119991041047323065577593)
(1703911033642285521501475481995/4, 1102343813623627845916169407198261090774939219/8)
(3652077712422208611593320763671/4, 5789523681738628554391589071533545434744586569/8)
(352026727500770729632317877760971/4, 6604701297779016008758035713310404474080024860031/8)
(1776743818924411871583988405023175/121, 74829595482338766924870171133618492376860560105646/1331)
(-5599939107511436010602903625091823/7921, 49159336464846881946892694393203461251199881716504/704969)
(32940950888329124567379444930867961/5041, 5953929118344485676637543868016722944517491049985298/357911)
(67874840820182268597647907936582871/502681, 80581032075217681677822192102437784338628376179256742/356400829)
(6713014538784551199121920448085296653/237169, 17389121945217432551592433873627102231449587231623719598/115501303)
(-316408858372840476395533341767285708/4748041, 3621148460240821165722126549633182690560042357051383059/10345981339)

Commentary

Specialization at T = 98513/165763 of the rank-16 fibration of X1092 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 7795441T^2 - 1152670T + 16097, p = 180186T - 120358, q = -440(10597T^2 + 11537T + 84), D = -96(15227912611T^3 + 5140797490T^2 - 7168552417T - 472316656), E = 16(131241922955T^4 + 380777938991T^3 - 82426481774T^2 - 23068210709T - 348132099).

last edited by wgxli at 2026-09-23 02:27:52 UTC · history