Trivial torsion group, rank ≥ 31

Alpöge - Howell (2026)

y2 + xy + y = x3 + x2 - 1284727764113567728281797636015784768866707681415849262157224232063x  
              + 560368321454261339256859338901915312332769858684945406858043869199456710681989058863306170127006181 

	Independent points of infinite order: 

P1 = [-343746913367434053557936715792581, -31005994945795167281668477280091616990476695982462]
P2 = [739387066521976456185309925500635, 3830813927243125109605210052773411116315836476962]
P3 = [794918160232596023344129483097739, 6435806665987309972375508567418781686192466975138]
P4 = [552229821725543064455253103966115, -4394327660511778936811414681478594663009127495638]
P5 = [621111741565922967463663441261795, 1421683191996488160880565213880200774834445145522]
P6 = [672709190481033101277453068263195, 739179814158401859498464392317452223287910224482]
P7 = [821399009280840681555588652225649, -7699942068198872209289633789793873375706107955662]
P8 = [765203485098820658379256355481245, -5034345308760298854369394902951465889284652680168]
P9 = [5399595603851507511243908479925915, 388654067273650918227195921742660550010310029109122]
P10 = [752668954339285868300409092384315, 4448429560131722871335630396999897920170009189762]
P11 = [665267573831105472345147950403835, -339632380726246052348178275348181915170728600638]
P12 = [2348702476455858806523697264250795, 102466166054736815527265002023727452468648322685682]
P13 = [-485301239340016399925702578908055, -32703997667874232639739344802658052943027850955518]
P14 = [1671562150318197588290891477490065, -55528472789639333688258829014227823359584879796238]
P15 = [533792265728561313885832952129951, -5165872676030535509137532850965718653070997396662]
P16 = [8050101189939018648004785213625795/9, -304090504548899861983493167851611965128494136836666/27]
P17 = [641126276756517817380587191947425, 475393761225519880788251703870437551199359754892]
P18 = [-28607312564278696457116932283279229/25, 2883492798944948226520935391601885431994599305099066/125]
P19 = [56549444812087071231733196839837255/4, 13405577698133557868624072930089297336164276021592851/8]
P20 = [1436747523773762064481823443661979, 40991896134294415817647888149334972652525806317538]
P21 = [-5064387018792502839844249474841045/9, 897570094862437631568619794896775730629375504169894/27]
P22 = [487104273264503086273608258117432335/484, -180314887995146178717423515157548125006198119570993099/10648]
P23 = [25599342024506626201702677512314955, 4091893764954118703319829997327215311497327967339042]
P24 = [980266989498097787981788009573917995/961, -525575897314714907955968902277850024344706090034709538/29791]
P25 = [5818507410725525932023407879789875/9, 1789182708478717961552943758166081786352797832214/27]
P26 = [19197683320190396803412613672365855/49, -3712724452088495818457900090458774587078999890353974/343]
P27 = [887913128901643530184527456121547975/841, -475948997371817062287059043149845391545859899478008302/24389] 
P28 = [1816718417495680977851120075801270195/1849, 1248545591609034704022269662335921779441905481482388694/79507]
P29 = [4528903098191425769774358894938128668457595/3630906049, -6558125371552387528524400913253424783234119422079260125534495614/218787505794593]
P30 = [33880100347989327554317013369507240405/52441, -1625428840401701290033088199518465040555146308891387272/12008989]
P31 = [-11477003798992551861310315487329257523535/8922169, -244939776343182008186576362205394985355490153552116110249914/26650518803]
Previous records with rank ≥ 30, rank ≥ 29 and rank ≥ 28.
Examples with rank ≥ 27, rank ≥ 26 and rank ≥ 25.
High rank curves with prescribed torsion Andrej Dujella home page