History of Kloosterman sums modulo a prime

 Display properties:   number-header: $K(a;p)$-Numbers: []+Numbers:+- params:+    p: '5'+    a: '1'+  number: '0.3819660112501051517954131656343618822796908201942371378645513772947395371810975502927927958106088625'+  comment: $\left(\frac{1}{5}\right)=1$, $[\mathbb{Q}(K(1;5)):\mathbb{Q}]=2$, a root+    of $x^2 - 3x + 1$+  equals: HREF{Algebraic_numbers_of_degree_2#1,-3,1,1}[$\frac{3-\sqrt{5}}{2}$]+- params:+    p: '5'+    a: '2'+  number: '-3.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275'+  comment: $\left(\frac{2}{5}\right)=-1$, $[\mathbb{Q}(K(2;5)):\mathbb{Q}]=2$, a root+    of $x^2 + 2x - 4$+  equals: HREF{Algebraic_numbers_of_degree_2#1,2,-4,1}[$-1-\sqrt{5}$]+- params:+    p: '5'+    a: '3'+  number: '1.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275'+  comment: $\left(\frac{3}{5}\right)=-1$, $[\mathbb{Q}(K(3;5)):\mathbb{Q}]=2$, a root+    of $x^2 + 2x - 4$+  equals: HREF{Algebraic_numbers_of_degree_2#1,2,-4,2}[$-1+\sqrt{5}$]+- params:+    p: '5'+    a: '4'+  number: '2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137'+  comment: $\left(\frac{4}{5}\right)=1$, $[\mathbb{Q}(K(4;5)):\mathbb{Q}]=2$, a root+    of $x^2 - 3x + 1$+  equals: HREF{Algebraic_numbers_of_degree_2#1,-3,1,2}[$\frac{3+\sqrt{5}}{2}$]+- params:+    p: '7'+    a: '1'+  number: '2.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574'+  comment: $\left(\frac{1}{7}\right)=1$, $[\mathbb{Q}(K(1;7)):\mathbb{Q}]=3$, a root+    of $x^3 + 3x^2 - 4x - 13$+- params:+    p: '7'+    a: '2'+  number: '-2.356895867892209443894399510021300583399127186734624389483150814604022376798785292853560804220878918'+  comment: $\left(\frac{2}{7}\right)=1$, $[\mathbb{Q}(K(2;7)):\mathbb{Q}]=3$, a root+    of $x^3 + 3x^2 - 4x - 13$+- params:+    p: '7'+    a: '3'+  number: '-1.603875471609676504944409278029780204663676648527428600214249692797729681711759862005445818874049661'+  comment: $\left(\frac{3}{7}\right)=-1$, $[\mathbb{Q}(K(3;7)):\mathbb{Q}]=3$, a root+    of $x^3 - 4x^2 - 4x + 8$+- params:+    p: '7'+    a: '4'+  number: '-2.692021471630095869627814897002069140197260599321894121355072909988549768970069207301047119869316656'+  comment: $\left(\frac{4}{7}\right)=1$, $[\mathbb{Q}(K(4;7)):\mathbb{Q}]=3$, a root+    of $x^3 + 3x^2 - 4x - 13$+- params:+    p: '7'+    a: '5'+  number: '4.493959207434934122100019536016959242529098923585608421462197756387414609825949138303770029306341487'+  comment: $\left(\frac{5}{7}\right)=-1$, $[\mathbb{Q}(K(5;7)):\mathbb{Q}]=3$, a root+    of $x^3 - 4x^2 - 4x + 8$+- params:+    p: '7'+    a: '6'+  number: '1.109916264174742382844389742012820962134577724941820178752051936410315071885810723701675789567708175'+  comment: $\left(\frac{6}{7}\right)=-1$, $[\mathbb{Q}(K(6;7)):\mathbb{Q}]=3$, a root+    of $x^3 - 4x^2 - 4x + 8$+- params:+    p: '11'+    a: '1'+  number: '-2.357872262870507966944322665871406480851360420922577399245066899502722756311168404533071251816641350'+  comment: $\left(\frac{1}{11}\right)=1$, $[\mathbb{Q}(K(1;11)):\mathbb{Q}]=5$, a+    root of $x^5 + 5x^4 - 12x^3 - 45x^2 + 5x + 23$+- params:+    p: '11'+    a: '2'+  number: '4.795754778231584113672075921211992194888964549178214280898269324490582469203074277365216142184451371'+  comment: $\left(\frac{2}{11}\right)=-1$, $[\mathbb{Q}(K(2;11)):\mathbb{Q}]=5$, a+    root of $x^5 - 6x^4 - 12x^3 + 120x^2 - 160x - 32$+- params:+    p: '11'+    a: '3'+  number: '0.7247241818887677836130196048920540363874398413515093365650111468534809777853569370029740258495484962'+  comment: $\left(\frac{3}{11}\right)=1$, $[\mathbb{Q}(K(3;11)):\mathbb{Q}]=5$, a+    root of $x^5 + 5x^4 - 12x^3 - 45x^2 + 5x + 23$+- params:+    p: '11'+    a: '4'+  number: '-5.716952715441700249450493047663376577781607236866441989062902591228573371487351267673488864766059080'+  comment: $\left(\frac{4}{11}\right)=1$, $[\mathbb{Q}(K(4;11)):\mathbb{Q}]=5$, a+    root of $x^5 + 5x^4 - 12x^3 - 45x^2 + 5x + 23$+- params:+    p: '11'+    a: '5'+  number: '-0.7575874396798351650018109698205792637787521222584648824435453383721677847803249827132360062626323350'+  comment: $\left(\frac{5}{11}\right)=1$, $[\mathbb{Q}(K(5;11)):\mathbb{Q}]=5$, a+    root of $x^5 + 5x^4 - 12x^3 - 45x^2 + 5x + 23$+- params:+    p: '11'+    a: '6'+  number: '-4.457414830239129815789172518130485008984984191036361530603162654004341592982566236250700709848308264'+  comment: $\left(\frac{6}{11}\right)=-1$, $[\mathbb{Q}(K(6;11)):\mathbb{Q}]=5$, a+    root of $x^5 - 6x^4 - 12x^3 + 120x^2 - 160x - 32$+- params:+    p: '11'+    a: '7'+  number: '3.092400698914405140341925922453014138931814197354209936032562183277737165375997143278354619169166418'+  comment: $\left(\frac{7}{11}\right)=-1$, $[\mathbb{Q}(K(7;11)):\mathbb{Q}]=5$, a+    root of $x^5 - 6x^4 - 12x^3 + 120x^2 - 160x - 32$+- params:+    p: '11'+    a: '8'+  number: '-0.1763118424504438574443756313468179821537997518305005704730869064670021094056547089965325550305720503'+  comment: $\left(\frac{8}{11}\right)=-1$, $[\mathbb{Q}(K(8;11)):\mathbb{Q}]=5$, a+    root of $x^5 - 6x^4 - 12x^3 + 120x^2 - 160x - 32$+- params:+    p: '11'+    a: '9'+  number: '3.107688236103275597783607078463308286024279938695974934186503682249982934793487717916822096995784268'+  comment: $\left(\frac{9}{11}\right)=1$, $[\mathbb{Q}(K(9;11)):\mathbb{Q}]=5$, a+    root of $x^5 + 5x^4 - 12x^3 - 45x^2 + 5x + 23$+- params:+    p: '11'+    a: '10'+  number: '2.745571195543584419219546305812296657318005196334437884145418052703024067809149524603662503525262526'+  comment: $\left(\frac{10}{11}\right)=-1$, $[\mathbb{Q}(K(10;11)):\mathbb{Q}]=5$,+    a root of $x^5 - 6x^4 - 12x^3 + 120x^2 - 160x - 32$+- params:+    p: '13'+    a: '1'+  number: '5.259534047905008684746565635915354866223557041841996932481982763295151986307151440192338189240818593'+  comment: $\left(\frac{1}{13}\right)=1$, $[\mathbb{Q}(K(1;13)):\mathbb{Q}]=6$, a+    root of $x^6 - 7x^5 - 11x^4 + 150x^3 - 232x^2 - 7x + 53$+- params:+    p: '13'+    a: '2'+  number: '-0.2396372847384889710950214124851647081571115139789462391720796542518217931847429490113112638047932163'+  comment: $\left(\frac{2}{13}\right)=-1$, $[\mathbb{Q}(K(2;13)):\mathbb{Q}]=6$, a+    root of $x^6 + 6x^5 - 24x^4 - 240x^3 - 544x^2 - 384x - 64$+- params:+    p: '13'+    a: '3'+  number: '-0.4332943234910524877515420199926023358636002718965456626384246569745266945507746571187989376294643427'+  comment: $\left(\frac{3}{13}\right)=1$, $[\mathbb{Q}(K(3;13)):\mathbb{Q}]=6$, a+    root of $x^6 - 7x^5 - 11x^4 + 150x^3 - 232x^2 - 7x + 53$+- params:+    p: '13'+    a: '4'+  number: '0.5690062201551475635694320525046240141154529848831786347966463123074790867919453658287195925077165126'+  comment: $\left(\frac{4}{13}\right)=1$, $[\mathbb{Q}(K(4;13)):\mathbb{Q}]=6$, a+    root of $x^6 - 7x^5 - 11x^4 + 150x^3 - 232x^2 - 7x + 53$+- params:+    p: '13'+    a: '5'+  number: '-3.029927830949727402459250439065164308130911941597120690118885574384881718785404934222760928243220018'+  comment: $\left(\frac{5}{13}\right)=-1$, $[\mathbb{Q}(K(5;13)):\mathbb{Q}]=6$, a+    root of $x^6 + 6x^5 - 24x^4 - 240x^3 - 544x^2 - 384x - 64$+- params:+    p: '13'+    a: '6'+  number: '-3.335986159775772919564949415920166929963273118269179283419487827590463436322862561970546889970477484'+  comment: $\left(\frac{6}{13}\right)=-1$, $[\mathbb{Q}(K(6;13)):\mathbb{Q}]=6$, a+    root of $x^6 + 6x^5 - 24x^4 - 240x^3 - 544x^2 - 384x - 64$+- params:+    p: '13'+    a: '7'+  number: '-0.8706384382417073148155718811450839182292344237683439427558671964293382011929776476341679665516000694'+  comment: $\left(\frac{7}{13}\right)=-1$, $[\mathbb{Q}(K(7;13)):\mathbb{Q}]=6$, a+    root of $x^6 + 6x^5 - 24x^4 - 240x^3 - 544x^2 - 384x - 64$+- params:+    p: '13'+    a: '8'+  number: '-4.820040096853058399110209925765056477170157514500907748849740284588030114881927712661208535268036223'+  comment: $\left(\frac{8}{13}\right)=-1$, $[\mathbb{Q}(K(8;13)):\mathbb{Q}]=6$, a+    root of $x^6 + 6x^5 - 24x^4 - 240x^3 - 544x^2 - 384x - 64$+- params:+    p: '13'+    a: '9'+  number: '4.082087188782027742683808285282991389016988090822418049222121478020125130683138884733389371416381827'+  comment: $\left(\frac{9}{13}\right)=1$, $[\mathbb{Q}(K(9;13)):\mathbb{Q}]=6$, a+    root of $x^6 - 7x^5 - 11x^4 + 150x^3 - 232x^2 - 7x + 53$+- params:+    p: '13'+    a: '10'+  number: '2.159403768526834893220052175365107346345506619553128127172879517815674995192192584238923169797934561'+  comment: $\left(\frac{10}{13}\right)=1$, $[\mathbb{Q}(K(10;13)):\mathbb{Q}]=6$,+    a root of $x^6 - 7x^5 - 11x^4 + 150x^3 - 232x^2 - 7x + 53$+- params:+    p: '13'+    a: '11'+  number: '6.296229810558755007045003074380636341650688512114497904316060537244535264367915805499995583838127010'+  comment: $\left(\frac{11}{13}\right)=-1$, $[\mathbb{Q}(K(11;13)):\mathbb{Q}]=6$,+    a root of $x^6 + 6x^5 - 24x^4 - 240x^3 - 544x^2 - 384x - 64$+- params:+    p: '13'+    a: '12'+  number: '-4.636736901877966396468316129075475279837904465204176081035205414463904504423653617874571385333387151'+  comment: $\left(\frac{12}{13}\right)=1$, $[\mathbb{Q}(K(12;13)):\mathbb{Q}]=6$,+    a root of $x^6 - 7x^5 - 11x^4 + 150x^3 - 232x^2 - 7x + 53$+- params:+    p: '17'+    a: '1'+  number: '-3.959065070099645892149391121370114865850325265546015074594919654254138857440298654172121657499140126'+  comment: $\left(\frac{1}{17}\right)=1$, $[\mathbb{Q}(K(1;17)):\mathbb{Q}]=8$, a+    root of $x^8 - 9x^7 - 23x^6 + 396x^5 - 695x^4 - 3004x^3 + 12268x^2 - 15530x ++    6733$+- params:+    p: '17'+    a: '2'+  number: '6.560736549697831787164415994752938902952745369360383269641248623052061081250314637818549486943333480'+  comment: $\left(\frac{2}{17}\right)=1$, $[\mathbb{Q}(K(2;17)):\mathbb{Q}]=8$, a+    root of $x^8 - 9x^7 - 23x^6 + 396x^5 - 695x^4 - 3004x^3 + 12268x^2 - 15530x ++    6733$+- params:+    p: '17'+    a: '3'+  number: '-1.701435252288094122627365453059430246402562564112869134104482935992612818774201371599606645742961194'+  comment: $\left(\frac{3}{17}\right)=-1$, $[\mathbb{Q}(K(3;17)):\mathbb{Q}]=8$, a+    root of $x^8 + 8x^7 - 40x^6 - 352x^5 + 240x^4 + 4544x^3 + 3904x^2 - 13184x - 17152$+- params:+    p: '17'+    a: '4'+  number: '1.418905133337239146078878348050604545394333686554441134386474019968164093074126792737966374791108417'+  comment: $\left(\frac{4}{17}\right)=1$, $[\mathbb{Q}(K(4;17)):\mathbb{Q}]=8$, a+    root of $x^8 - 9x^7 - 23x^6 + 396x^5 - 695x^4 - 3004x^3 + 12268x^2 - 15530x ++    6733$+- params:+    p: '17'+    a: '5'+  number: '4.644459478836242106234852615447509403066232398642494776978181432725555651057005327394434248538171243'+  comment: $\left(\frac{5}{17}\right)=-1$, $[\mathbb{Q}(K(5;17)):\mathbb{Q}]=8$, a+    root of $x^8 + 8x^7 - 40x^6 - 352x^5 + 240x^4 + 4544x^3 + 3904x^2 - 13184x - 17152$+- params:+    p: '17'+    a: '6'+  number: '-2.343493642247427012148452807408688617219271763356669940222489476225373944763040635149303718014638197'+  comment: $\left(\frac{6}{17}\right)=-1$, $[\mathbb{Q}(K(6;17)):\mathbb{Q}]=8$, a+    root of $x^8 + 8x^7 - 40x^6 - 352x^5 + 240x^4 + 4544x^3 + 3904x^2 - 13184x - 17152$+- params:+    p: '17'+    a: '7'+  number: '-7.960346064064936208792893493647618163042680047930130082878438948265162777713353490631430150176175119'+  comment: $\left(\frac{7}{17}\right)=-1$, $[\mathbb{Q}(K(7;17)):\mathbb{Q}]=8$, a+    root of $x^8 + 8x^7 - 40x^6 - 352x^5 + 240x^4 + 4544x^3 + 3904x^2 - 13184x - 17152$+- params:+    p: '17'+    a: '8'+  number: '-5.673791777858652993151831416648835546360001349748721027373473007151527350820432252285691367532102220'+  comment: $\left(\frac{8}{17}\right)=1$, $[\mathbb{Q}(K(8;17)):\mathbb{Q}]=8$, a+    root of $x^8 - 9x^7 - 23x^6 + 396x^5 - 695x^4 - 3004x^3 + 12268x^2 - 15530x ++    6733$+- params:+    p: '17'+    a: '9'+  number: '4.079786770823090295096848805005756490969451112084378635436703675962688145303113934461768011465277119'+  comment: $\left(\frac{9}{17}\right)=1$, $[\mathbb{Q}(K(9;17)):\mathbb{Q}]=8$, a+    root of $x^8 - 9x^7 - 23x^6 + 396x^5 - 695x^4 - 3004x^3 + 12268x^2 - 15530x ++    6733$+- params:+    p: '17'+    a: '10'+  number: '-3.950023379841177155709102814001432023818239840898939513771399821717394072553505598681785958483333353'+  comment: $\left(\frac{10}{17}\right)=-1$, $[\mathbb{Q}(K(10;17)):\mathbb{Q}]=8$,+    a root of $x^8 + 8x^7 - 40x^6 - 352x^5 + 240x^4 + 4544x^3 + 3904x^2 - 13184x -+    17152$+- params:+    p: '17'+    a: '11'+  number: '2.007651834918219277007629403109584753785793201438498668075061100018022102354656537286792525052778076'+  comment: $\left(\frac{11}{17}\right)=-1$, $[\mathbb{Q}(K(11;17)):\mathbb{Q}]=8$,+    a root of $x^8 + 8x^7 - 40x^6 - 352x^5 + 240x^4 + 4544x^3 + 3904x^2 - 13184x -+    17152$+- params:+    p: '17'+    a: '12'+  number: '-3.233798586183432522413303075313923631775112568490421847710246295359769190259111193042403831571324337'+  comment: $\left(\frac{12}{17}\right)=-1$, $[\mathbb{Q}(K(12;17)):\mathbb{Q}]=8$,+    a root of $x^8 + 8x^7 - 40x^6 - 352x^5 + 240x^4 + 4544x^3 + 3904x^2 - 13184x -+    17152$+- params:+    p: '17'+    a: '13'+  number: '3.606432774251047102976313789831737629212097458277579999854247657626923284133600336232207130509960597'+  comment: $\left(\frac{13}{17}\right)=1$, $[\mathbb{Q}(K(13;17)):\mathbb{Q}]=8$,+    a root of $x^8 - 9x^7 - 23x^6 + 396x^5 - 695x^4 - 3004x^3 + 12268x^2 - 15530x+    + 6733$+- params:+    p: '17'+    a: '14'+  number: '4.536985610870605638448635624873998525405841184708037073633814944816735050651550424423303530397482881'+  comment: $\left(\frac{14}{17}\right)=-1$, $[\mathbb{Q}(K(14;17)):\mathbb{Q}]=8$,+    a root of $x^8 + 8x^7 - 40x^6 - 352x^5 + 240x^4 + 4544x^3 + 3904x^2 - 13184x -+    17152$+- params:+    p: '17'+    a: '15'+  number: '1.594821270146561185801271544877178665011404480990769339494837494684255297435814476799305694528833769'+  comment: $\left(\frac{15}{17}\right)=1$, $[\mathbb{Q}(K(15;17)):\mathbb{Q}]=8$,+    a root of $x^8 - 9x^7 - 23x^6 + 396x^5 - 695x^4 - 3004x^3 + 12268x^2 - 15530x+    + 6733$+- params:+    p: '17'+    a: '16'+  number: '1.372174349702529368183494055500734178670294508027183723154881190111574307063760728408016326792728964'+  comment: $\left(\frac{16}{17}\right)=1$, $[\mathbb{Q}(K(16;17)):\mathbb{Q}]=8$,+    a root of $x^8 - 9x^7 - 23x^6 + 396x^5 - 695x^4 - 3004x^3 + 12268x^2 - 15530x+    + 6733$+- params:+    p: '19'+    a: '1'+  number: '0.8947711179711796199050354254639302787001572420274377409316915010461418157802867488423246531070776214'+  comment: $\left(\frac{1}{19}\right)=1$, $[\mathbb{Q}(K(1;19)):\mathbb{Q}]=9$, a+    root of $x^9 + 9x^8 - 40x^7 - 467x^6 + 145x^5 + 6871x^4 + 4986x^3 - 29262x^2 -+    19447x + 33023$+- params:+    p: '19'+    a: '2'+  number: '-1.694751661622389124031728168411743963619660632021813056570267216348319751003766503805851764683419510'+  comment: $\left(\frac{2}{19}\right)=-1$, $[\mathbb{Q}(K(2;19)):\mathbb{Q}]=9$, a+    root of $x^9 - 10x^8 - 40x^7 + 464x^6 + 848x^5 - 6144x^4 - 13824x^3 + 10752x^2+    + 39168x + 18944$+- params:+    p: '19'+    a: '3'+  number: '6.719514920026600661344772616596130584906651538985027457778602415035962700550185846102452325849114443'+  comment: $\left(\frac{3}{19}\right)=-1$, $[\mathbb{Q}(K(3;19)):\mathbb{Q}]=9$, a+    root of $x^9 - 10x^8 - 40x^7 + 464x^6 + 848x^5 - 6144x^4 - 13824x^3 + 10752x^2+    + 39168x + 18944$+- params:+    p: '19'+    a: '4'+  number: '5.493702898168504282330426914408640673748369442578659986148109495691599653672512777251320233335670701'+  comment: $\left(\frac{4}{19}\right)=1$, $[\mathbb{Q}(K(4;19)):\mathbb{Q}]=9$, a+    root of $x^9 + 9x^8 - 40x^7 - 467x^6 + 145x^5 + 6871x^4 + 4986x^3 - 29262x^2 -+    19447x + 33023$+- params:+    p: '19'+    a: '5'+  number: '-7.508804815536783550402866746153979375919138407052771016200926132491638446859939058849535967103890594'+  comment: $\left(\frac{5}{19}\right)=1$, $[\mathbb{Q}(K(5;19)):\mathbb{Q}]=9$, a+    root of $x^9 + 9x^8 - 40x^7 - 467x^6 + 145x^5 + 6871x^4 + 4986x^3 - 29262x^2 -+    19447x + 33023$+- params:+    p: '19'+    a: '6'+  number: '-1.842431694144834250118783406600161781710528297076572413123935922923923662321801542958895020270120914'+  comment: $\left(\frac{6}{19}\right)=1$, $[\mathbb{Q}(K(6;19)):\mathbb{Q}]=9$, a+    root of $x^9 + 9x^8 - 40x^7 - 467x^6 + 145x^5 + 6871x^4 + 4986x^3 - 29262x^2 -+    19447x + 33023$+- params:+    p: '19'+    a: '7'+  number: '2.010894844895303477226557265240499956314430693047467126252383511185391347010182794311304604675964339'+  comment: $\left(\frac{7}{19}\right)=1$, $[\mathbb{Q}(K(7;19)):\mathbb{Q}]=9$, a+    root of $x^9 + 9x^8 - 40x^7 - 467x^6 + 145x^5 + 6871x^4 + 4986x^3 - 29262x^2 -+    19447x + 33023$+- params:+    p: '19'+    a: '8'+  number: '-2.478084231923192907447344499089473814307699519836114802498665190210841010263790756646624105949709352'+  comment: $\left(\frac{8}{19}\right)=-1$, $[\mathbb{Q}(K(8;19)):\mathbb{Q}]=9$, a+    root of $x^9 - 10x^8 - 40x^7 + 464x^6 + 848x^5 - 6144x^4 - 13824x^3 + 10752x^2+    + 39168x + 18944$+- params:+    p: '19'+    a: '9'+  number: '-2.922376397667502799284163533907750247412669230963965164296676285923713783688194341844222634801030383'+  comment: $\left(\frac{9}{19}\right)=1$, $[\mathbb{Q}(K(9;19)):\mathbb{Q}]=9$, a+    root of $x^9 + 9x^8 - 40x^7 - 467x^6 + 145x^5 + 6871x^4 + 4986x^3 - 29262x^2 -+    19447x + 33023$+- params:+    p: '19'+    a: '10'+  number: '-2.369546040728542384166347707674874254172431085801747147903712544553236299209726782706293078598220990'+  comment: $\left(\frac{10}{19}\right)=-1$, $[\mathbb{Q}(K(10;19)):\mathbb{Q}]=9$,+    a root of $x^9 - 10x^8 - 40x^7 + 464x^6 + 848x^5 - 6144x^4 - 13824x^3 + 10752x^2+    + 39168x + 18944$+- params:+    p: '19'+    a: '11'+  number: '-4.842399643299888667004152867358853437187002503760040659259323050860526683652443413322024188453853554'+  comment: $\left(\frac{11}{19}\right)=1$, $[\mathbb{Q}(K(11;19)):\mathbb{Q}]=9$,+    a root of $x^9 + 9x^8 - 40x^7 - 467x^6 + 145x^5 + 6871x^4 + 4986x^3 - 29262x^2+    - 19447x + 33023$+- params:+    p: '19'+    a: '12'+  number: '-4.851861041377601818990225925294853726433274381212048000717756930029621095395931703451231952109055555'+  comment: $\left(\frac{12}{19}\right)=-1$, $[\mathbb{Q}(K(12;19)):\mathbb{Q}]=9$,+    a root of $x^9 - 10x^8 - 40x^7 + 464x^6 + 848x^5 - 6144x^4 - 13824x^3 + 10752x^2+    + 39168x + 18944$+- params:+    p: '19'+    a: '13'+  number: '5.900387335995819515504607779803865296569579415088420880722654725043256721492102092467092477710216555'+  comment: $\left(\frac{13}{19}\right)=-1$, $[\mathbb{Q}(K(13;19)):\mathbb{Q}]=9$,+    a root of $x^9 - 10x^8 - 40x^7 + 464x^6 + 848x^5 - 6144x^4 - 13824x^3 + 10752x^2+    + 39168x + 18944$+- params:+    p: '19'+    a: '14'+  number: '1.862741390648552199753675752555815841109791070667796533195325235276298021819568341192196117414746396'+  comment: $\left(\frac{14}{19}\right)=-1$, $[\mathbb{Q}(K(14;19)):\mathbb{Q}]=9$,+    a root of $x^9 - 10x^8 - 40x^7 + 464x^6 + 848x^5 - 6144x^4 - 13824x^3 + 10752x^2+    + 39168x + 18944$+- params:+    p: '19'+    a: '15'+  number: '7.609728632051864326238219421993518611309975373603794033283726098957946465635716893054417556785199392'+  comment: $\left(\frac{15}{19}\right)=-1$, $[\mathbb{Q}(K(15;19)):\mathbb{Q}]=9$,+    a root of $x^9 - 10x^8 - 40x^7 + 464x^6 + 848x^5 - 6144x^4 - 13824x^3 + 10752x^2+    + 39168x + 18944$+- params:+    p: '19'+    a: '16'+  number: '-4.275015016378060943675261958240608356173733155663618148838244922920146524390978662143763155838606996'+  comment: $\left(\frac{16}{19}\right)=1$, $[\mathbb{Q}(K(16;19)):\mathbb{Q}]=9$,+    a root of $x^9 + 9x^8 - 40x^7 - 467x^6 + 145x^5 + 6871x^4 + 4986x^3 - 29262x^2+    - 19447x + 33023$+- params:+    p: '19'+    a: '17'+  number: '3.991658705992082831023208907148282289640114216863402548386921807196816284450374698713491475348789779'+  comment: $\left(\frac{17}{19}\right)=1$, $[\mathbb{Q}(K(17;19)):\mathbb{Q}]=9$,+    a root of $x^9 + 9x^8 - 40x^7 - 467x^6 + 145x^5 + 6871x^4 + 4986x^3 - 29262x^2+    - 19447x + 33023$+- params:+    p: '19'+    a: '18'+  number: '-0.6981293030711104682056292704783845753629317794733158972899065931714457536243574262061575764188713798'+  comment: $\left(\frac{18}{19}\right)=-1$, $[\mathbb{Q}(K(18;19)):\mathbb{Q}]=9$,+    a root of $x^9 - 10x^8 - 40x^7 + 464x^6 + 848x^5 - 6144x^4 - 13824x^3 + 10752x^2+    + 39168x + 18944$+- params:+    p: '23'+    a: '1'+  number: '-7.458304846425912603330937313870960891077013412896397990296378125735808108380026482024581830910015294'+  comment: $\left(\frac{1}{23}\right)=1$, $[\mathbb{Q}(K(1;23)):\mathbb{Q}]=11$, a+    root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 - 103101x^4+    + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '2'+  number: '2.145355658677892144040872387025985492285699501841777464208824717704832544023679114355236475571057118'+  comment: $\left(\frac{2}{23}\right)=1$, $[\mathbb{Q}(K(2;23)):\mathbb{Q}]=11$, a+    root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 - 103101x^4+    + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '3'+  number: '2.374639829199348760968163277886754903939797276616115273656928001085821309594083890899685794135996319'+  comment: $\left(\frac{3}{23}\right)=1$, $[\mathbb{Q}(K(3;23)):\mathbb{Q}]=11$, a+    root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 - 103101x^4+    + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '4'+  number: '2.046370777333861348780826037399099491566842305407847281217323828014821553147351593356665582091052975'+  comment: $\left(\frac{4}{23}\right)=1$, $[\mathbb{Q}(K(4;23)):\mathbb{Q}]=11$, a+    root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 - 103101x^4+    + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '5'+  number: '3.006649835977861990672538531116896763493109958715678636339138462072247496948669883607902470487700614'+  comment: $\left(\frac{5}{23}\right)=-1$, $[\mathbb{Q}(K(5;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '23'+    a: '6'+  number: '7.275060715824075779716360543202560092645826983528240877076266983329270967274843630020597234452235735'+  comment: $\left(\frac{6}{23}\right)=1$, $[\mathbb{Q}(K(6;23)):\mathbb{Q}]=11$, a+    root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 - 103101x^4+    + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '7'+  number: '-5.481067359210671217935310467914371074573962024241869676823660420453555444308479401649371613397041528'+  comment: $\left(\frac{7}{23}\right)=-1$, $[\mathbb{Q}(K(7;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '23'+    a: '8'+  number: '-0.08592814790203416139929860384299691049846383031398706229236193084208672960584967936905600870587034978'+  comment: $\left(\frac{8}{23}\right)=1$, $[\mathbb{Q}(K(8;23)):\mathbb{Q}]=11$, a+    root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 - 103101x^4+    + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '9'+  number: '-6.453508062445277976740932946193643783873999952438065434664473512417531160679478675181737515225411239'+  comment: $\left(\frac{9}{23}\right)=1$, $[\mathbb{Q}(K(9;23)):\mathbb{Q}]=11$, a+    root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 - 103101x^4+    + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '10'+  number: '-7.898004631895880250858234380475629821744742373464376519783464006737639097111439963792712146237163901'+  comment: $\left(\frac{10}{23}\right)=-1$, $[\mathbb{Q}(K(10;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '23'+    a: '11'+  number: '7.960687186566972044562882954799255266394540173763844837543665073488308672351776066044584572159898884'+  comment: $\left(\frac{11}{23}\right)=-1$, $[\mathbb{Q}(K(11;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '23'+    a: '12'+  number: '-7.453239083019326463632247094382097867519272935214960891266007184910256482563528563443956492930900726'+  comment: $\left(\frac{12}{23}\right)=1$, $[\mathbb{Q}(K(12;23)):\mathbb{Q}]=11$,+    a root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 -+    103101x^4 + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '13'+  number: '-4.477901827105807857669420286017315353492414539459032527029458324523528935040671686276877894312832253'+  comment: $\left(\frac{13}{23}\right)=1$, $[\mathbb{Q}(K(13;23)):\mathbb{Q}]=11$,+    a root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 -+    103101x^4 + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '14'+  number: '-3.181794402495167800519508305183148421796740645931110660743907991437939467064132189270122340800648718'+  comment: $\left(\frac{14}{23}\right)=-1$, $[\mathbb{Q}(K(14;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '23'+    a: '15'+  number: '6.341910604841765873020662314476000768873022050982208555086241683072570505005861369270985234364281805'+  comment: $\left(\frac{15}{23}\right)=-1$, $[\mathbb{Q}(K(15;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '23'+    a: '16'+  number: '0.8346528151829765248592243258163940491305168021741824501703269665592214764001021908376047365489364750'+  comment: $\left(\frac{16}{23}\right)=1$, $[\mathbb{Q}(K(16;23)):\mathbb{Q}]=11$,+    a root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 -+    103101x^4 + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '17'+  number: '-1.445715615921703567799076841392901282610343072333100137988442948475914815365846669591660130568747192'+  comment: $\left(\frac{17}{23}\right)=-1$, $[\mathbb{Q}(K(17;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '23'+    a: '18'+  number: '0.2528021706802045044073896729762207768924818007542805592190085817352435658294946668264199192857512395'+  comment: $\left(\frac{18}{23}\right)=1$, $[\mathbb{Q}(K(18;23)):\mathbb{Q}]=11$,+    a root of $x^{11} + 11x^{10} - 60x^9 - 847x^8 + 606x^7 + 16010x^6 - 16558x^5 -+    103101x^4 + 227819x^3 - 134449x^2 + 12408x + 2209$+- params:+    p: '23'+    a: '19'+  number: '3.442481916529264020116592401027985131244580669151808598337933073312019807500604778870652248957399418'+  comment: $\left(\frac{19}{23}\right)=-1$, $[\mathbb{Q}(K(19;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '23'+    a: '20'+  number: '1.930788167155465249035307067341150698968242509882574476834596470610610639309656186066006103578224808'+  comment: $\left(\frac{20}{23}\right)=-1$, $[\mathbb{Q}(K(20;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '23'+    a: '21'+  number: '5.134040424020479258040645014007892309166520127312959907533795258561609598134008996643272541868815113'+  comment: $\left(\frac{21}{23}\right)=-1$, $[\mathbb{Q}(K(21;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '23'+    a: '22'+  number: '2.190023874431614401663501712196869662585772626161381983664105345987682104599320943800463059587280697'+  comment: $\left(\frac{22}{23}\right)=-1$, $[\mathbb{Q}(K(22;23)):\mathbb{Q}]=11$,+    a root of $x^{11} - 12x^{10} - 60x^9 + 1200x^8 - 1648x^7 - 30496x^6 + 110080x^5+    + 131200x^4 - 1063424x^3 + 901632x^2 + 1718272x - 2258944$+- params:+    p: '29'+    a: '1'+  number: '-5.150692882037460468989164694719719531599310721287578347655257338137039143105342164916749971054479821'+  comment: $\left(\frac{1}{29}\right)=1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '2'+  number: '2.328479867612536983816280849797787716198992458077042882820029483067799074392650098547148088176956304'+  comment: $\left(\frac{2}{29}\right)=-1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '3'+  number: '-3.689994663980114395532345719699887626117049362057293297673691381502435718865072181705125820161021549'+  comment: $\left(\frac{3}{29}\right)=-1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '4'+  number: '4.182343133666848345321588542062490966744707435754273794762002020606966060415942415443090006749038082'+  comment: $\left(\frac{4}{29}\right)=1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '5'+  number: '5.236510805717278410974786105159971009763542985668385394338130539382314392390184212832373911947917652'+  comment: $\left(\frac{5}{29}\right)=1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '6'+  number: '4.823685630269386314681912110097272892746643755967731937528787821914264496671671766966121016014581667'+  comment: $\left(\frac{6}{29}\right)=1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '7'+  number: '-8.882239762263238253669037378619915303307437867089875796777737567242765566186514592508879273917331551'+  comment: $\left(\frac{7}{29}\right)=1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '8'+  number: '7.439984871519743191294825322547168446518491373238835388839810213173018512639874117781243317905958334'+  comment: $\left(\frac{8}{29}\right)=-1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '9'+  number: '4.875210804347523225835711461420336037371091592260545088794293659484733168108778607879513397550537352'+  comment: $\left(\frac{9}{29}\right)=1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '10'+  number: '0.1647931676349000447590279209460428907720932906300043697458921403390033030534825217497695507709631531'+  comment: $\left(\frac{10}{29}\right)=-1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '11'+  number: '-5.851248167938508451867994549687577146544802930588535291845855043214630017803908862610036730980001728'+  comment: $\left(\frac{11}{29}\right)=-1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '12'+  number: '-9.500280861280981904436456896310694138054650556168645442677980738582855149758223165027246528266013232'+  comment: $\left(\frac{12}{29}\right)=-1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '13'+  number: '-2.368728812614293330399437244865269776612758921573968943000660491798458730437704683109783692767826112'+  comment: $\left(\frac{13}{29}\right)=1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '14'+  number: '-6.871544455197594670063996052168578154951102961688569814816813232200223070391467485250320604895904896'+  comment: $\left(\frac{14}{29}\right)=-1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '15'+  number: '1.620309103019258439374413186533588985141689912977803768695080338222108303434143410113994103917244081'+  comment: $\left(\frac{15}{29}\right)=-1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '16'+  number: '-5.899679178578147759023723836376212358805982568928098976358946693199197336784837212993181335622467898'+  comment: $\left(\frac{16}{29}\right)=1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '17'+  number: '8.279038728210820721846200010889191960969262001706660239961055452869774085521557940896660840602092724'+  comment: $\left(\frac{17}{29}\right)=-1$, degree $14$ over $\mathbb{Q}$+- params:+    p: '29'+    a: '18'+  number: '1.748240623619764769114328982913544231197922320788718905378326749783205970169786387022296652813410942'+  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comment: $\left(\frac{44}{71}\right)=-1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '45'+  number: '3.614201194352286144592767833025525073810970513496999859507064645693639631791462358894877874615877160'+  comment: $\left(\frac{45}{71}\right)=1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '46'+  number: '0.2305431725462653009077157594236400374381153764406044611333835226980637409976897392780487032889380391'+  comment: $\left(\frac{46}{71}\right)=-1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '47'+  number: '5.316932591776642023821888278422723439879684391578373378637375178384054095826176945284236926993645414'+  comment: $\left(\frac{47}{71}\right)=-1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '48'+  number: '1.064540845024287898197420594771110570779171620643319181411441119647901697101701443377097353166927111'+  comment: $\left(\frac{48}{71}\right)=1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '49'+  number: '-11.03896610714770323624467943925063224300944580342347403183076816193885731066662437343446508279015609'+  comment: $\left(\frac{49}{71}\right)=1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '50'+  number: '-4.902427704642280585496874916880591675630022392252501235810901486601391153779705175567335687793703247'+  comment: $\left(\frac{50}{71}\right)=1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '51'+  number: '13.20043402939398758860654859113736842373646823625128381454036284448061703411338787740967293687160329'+  comment: $\left(\frac{51}{71}\right)=-1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '52'+  number: '0.6725000969668083552016788988628273230752969191052718510292660587339995803092123442748795096005861063'+  comment: $\left(\frac{52}{71}\right)=-1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '53'+  number: 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number: '8.862971038904968521776809924017272499980941207355344483749438686741760736540327389989435553347234466'+  comment: $\left(\frac{62}{71}\right)=-1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '63'+  number: '5.175773130664318975379732472731219009800523381235955706929363744020109960510590946758768054115239348'+  comment: $\left(\frac{63}{71}\right)=-1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '64'+  number: '8.440472435227801352538755953989708340967020570434254527234214394639201971861036828141823510821523196'+  comment: $\left(\frac{64}{71}\right)=1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '65'+  number: '1.191643123402609780484739858255614751357836275664909325911041220932675961569524012162615198463227160'+  comment: $\left(\frac{65}{71}\right)=-1$, degree $35$ over $\mathbb{Q}$+- params:+    p: '71'+    a: '66'+  number: '1.062815136924648385018006393006452257305539643704113829833534552804852644060321642861768080151352625'+  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