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Properties

Label 40.40.6306559910...0625.1
Degree $40$
Signature $[40, 0]$
Discriminant $5^{64}\cdot 17^{35}$
Root discriminant $156.67$
Ramified primes $5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{40}$ (as 40T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-838099, -12223285, 32750635, 788314970, 1075428145, -9524536784, -20491798285, 44436502095, 128198823150, -88397056585, -400803896176, 41301908330, 729886520165, 131445427210, -855026205860, -284987100906, 687800308605, 288115234125, -396720671620, -183419877685, 168812732530, 80258980940, -53940046905, -25070547550, 13060739610, 5686194526, -2400623515, -940576570, 333224725, 112830010, -34503692, -9655440, 2609590, 571320, -139325, -22094, 4955, 500, -105, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 5*x^39 - 105*x^38 + 500*x^37 + 4955*x^36 - 22094*x^35 - 139325*x^34 + 571320*x^33 + 2609590*x^32 - 9655440*x^31 - 34503692*x^30 + 112830010*x^29 + 333224725*x^28 - 940576570*x^27 - 2400623515*x^26 + 5686194526*x^25 + 13060739610*x^24 - 25070547550*x^23 - 53940046905*x^22 + 80258980940*x^21 + 168812732530*x^20 - 183419877685*x^19 - 396720671620*x^18 + 288115234125*x^17 + 687800308605*x^16 - 284987100906*x^15 - 855026205860*x^14 + 131445427210*x^13 + 729886520165*x^12 + 41301908330*x^11 - 400803896176*x^10 - 88397056585*x^9 + 128198823150*x^8 + 44436502095*x^7 - 20491798285*x^6 - 9524536784*x^5 + 1075428145*x^4 + 788314970*x^3 + 32750635*x^2 - 12223285*x - 838099)
 
gp: K = bnfinit(x^40 - 5*x^39 - 105*x^38 + 500*x^37 + 4955*x^36 - 22094*x^35 - 139325*x^34 + 571320*x^33 + 2609590*x^32 - 9655440*x^31 - 34503692*x^30 + 112830010*x^29 + 333224725*x^28 - 940576570*x^27 - 2400623515*x^26 + 5686194526*x^25 + 13060739610*x^24 - 25070547550*x^23 - 53940046905*x^22 + 80258980940*x^21 + 168812732530*x^20 - 183419877685*x^19 - 396720671620*x^18 + 288115234125*x^17 + 687800308605*x^16 - 284987100906*x^15 - 855026205860*x^14 + 131445427210*x^13 + 729886520165*x^12 + 41301908330*x^11 - 400803896176*x^10 - 88397056585*x^9 + 128198823150*x^8 + 44436502095*x^7 - 20491798285*x^6 - 9524536784*x^5 + 1075428145*x^4 + 788314970*x^3 + 32750635*x^2 - 12223285*x - 838099, 1)
 

Normalized defining polynomial

\( x^{40} - 5 x^{39} - 105 x^{38} + 500 x^{37} + 4955 x^{36} - 22094 x^{35} - 139325 x^{34} + 571320 x^{33} + 2609590 x^{32} - 9655440 x^{31} - 34503692 x^{30} + 112830010 x^{29} + 333224725 x^{28} - 940576570 x^{27} - 2400623515 x^{26} + 5686194526 x^{25} + 13060739610 x^{24} - 25070547550 x^{23} - 53940046905 x^{22} + 80258980940 x^{21} + 168812732530 x^{20} - 183419877685 x^{19} - 396720671620 x^{18} + 288115234125 x^{17} + 687800308605 x^{16} - 284987100906 x^{15} - 855026205860 x^{14} + 131445427210 x^{13} + 729886520165 x^{12} + 41301908330 x^{11} - 400803896176 x^{10} - 88397056585 x^{9} + 128198823150 x^{8} + 44436502095 x^{7} - 20491798285 x^{6} - 9524536784 x^{5} + 1075428145 x^{4} + 788314970 x^{3} + 32750635 x^{2} - 12223285 x - 838099 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[40, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6306559910287043207279387718742999783337498304926216352583878688164986670017242431640625=5^{64}\cdot 17^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.67$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(425=5^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{425}(256,·)$, $\chi_{425}(1,·)$, $\chi_{425}(271,·)$, $\chi_{425}(16,·)$, $\chi_{425}(276,·)$, $\chi_{425}(21,·)$, $\chi_{425}(406,·)$, $\chi_{425}(151,·)$, $\chi_{425}(281,·)$, $\chi_{425}(26,·)$, $\chi_{425}(416,·)$, $\chi_{425}(161,·)$, $\chi_{425}(291,·)$, $\chi_{425}(36,·)$, $\chi_{425}(421,·)$, $\chi_{425}(166,·)$, $\chi_{425}(171,·)$, $\chi_{425}(186,·)$, $\chi_{425}(191,·)$, $\chi_{425}(321,·)$, $\chi_{425}(66,·)$, $\chi_{425}(196,·)$, $\chi_{425}(331,·)$, $\chi_{425}(76,·)$, $\chi_{425}(206,·)$, $\chi_{425}(336,·)$, $\chi_{425}(81,·)$, $\chi_{425}(341,·)$, $\chi_{425}(86,·)$, $\chi_{425}(356,·)$, $\chi_{425}(101,·)$, $\chi_{425}(361,·)$, $\chi_{425}(106,·)$, $\chi_{425}(236,·)$, $\chi_{425}(366,·)$, $\chi_{425}(111,·)$, $\chi_{425}(246,·)$, $\chi_{425}(376,·)$, $\chi_{425}(121,·)$, $\chi_{425}(251,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{7} a^{32} - \frac{3}{7} a^{29} - \frac{2}{7} a^{28} - \frac{1}{7} a^{27} + \frac{3}{7} a^{26} - \frac{3}{7} a^{25} + \frac{3}{7} a^{24} + \frac{3}{7} a^{23} + \frac{2}{7} a^{22} - \frac{3}{7} a^{21} - \frac{1}{7} a^{20} + \frac{2}{7} a^{19} - \frac{3}{7} a^{18} + \frac{1}{7} a^{17} + \frac{1}{7} a^{16} + \frac{1}{7} a^{15} + \frac{3}{7} a^{14} + \frac{3}{7} a^{13} - \frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{33} - \frac{3}{7} a^{30} - \frac{2}{7} a^{29} - \frac{1}{7} a^{28} + \frac{3}{7} a^{27} - \frac{3}{7} a^{26} + \frac{3}{7} a^{25} + \frac{3}{7} a^{24} + \frac{2}{7} a^{23} - \frac{3}{7} a^{22} - \frac{1}{7} a^{21} + \frac{2}{7} a^{20} - \frac{3}{7} a^{19} + \frac{1}{7} a^{18} + \frac{1}{7} a^{17} + \frac{1}{7} a^{16} + \frac{3}{7} a^{15} + \frac{3}{7} a^{14} - \frac{1}{7} a^{13} + \frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{1}{7} a^{10} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{34} - \frac{3}{7} a^{31} - \frac{2}{7} a^{30} - \frac{1}{7} a^{29} + \frac{3}{7} a^{28} - \frac{3}{7} a^{27} + \frac{3}{7} a^{26} + \frac{3}{7} a^{25} + \frac{2}{7} a^{24} - \frac{3}{7} a^{23} - \frac{1}{7} a^{22} + \frac{2}{7} a^{21} - \frac{3}{7} a^{20} + \frac{1}{7} a^{19} + \frac{1}{7} a^{18} + \frac{1}{7} a^{17} + \frac{3}{7} a^{16} + \frac{3}{7} a^{15} - \frac{1}{7} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{707} a^{35} - \frac{19}{707} a^{34} - \frac{10}{707} a^{33} - \frac{50}{707} a^{32} - \frac{239}{707} a^{31} + \frac{347}{707} a^{30} - \frac{111}{707} a^{29} + \frac{240}{707} a^{28} + \frac{33}{101} a^{27} + \frac{346}{707} a^{26} - \frac{30}{101} a^{25} - \frac{30}{707} a^{24} - \frac{26}{101} a^{23} + \frac{195}{707} a^{22} - \frac{240}{707} a^{21} - \frac{118}{707} a^{20} + \frac{3}{7} a^{19} - \frac{27}{707} a^{18} - \frac{136}{707} a^{17} - \frac{104}{707} a^{16} + \frac{89}{707} a^{15} - \frac{144}{707} a^{14} - \frac{214}{707} a^{13} + \frac{253}{707} a^{12} - \frac{37}{101} a^{11} + \frac{30}{101} a^{10} + \frac{34}{101} a^{9} + \frac{79}{707} a^{8} + \frac{50}{707} a^{7} - \frac{173}{707} a^{6} - \frac{211}{707} a^{5} + \frac{167}{707} a^{4} - \frac{249}{707} a^{3} - \frac{76}{707} a^{2} + \frac{33}{707} a + \frac{323}{707}$, $\frac{1}{707} a^{36} + \frac{33}{707} a^{34} - \frac{38}{707} a^{33} + \frac{23}{707} a^{32} + \frac{250}{707} a^{31} + \frac{17}{101} a^{30} + \frac{50}{707} a^{29} - \frac{158}{707} a^{28} + \frac{89}{707} a^{27} + \frac{1}{707} a^{26} - \frac{26}{101} a^{25} + \frac{8}{101} a^{24} + \frac{272}{707} a^{23} - \frac{10}{101} a^{22} + \frac{69}{707} a^{21} + \frac{283}{707} a^{20} + \frac{25}{101} a^{19} - \frac{144}{707} a^{18} - \frac{163}{707} a^{17} + \frac{32}{707} a^{16} + \frac{335}{707} a^{15} + \frac{181}{707} a^{14} + \frac{25}{707} a^{13} + \frac{306}{707} a^{12} + \frac{34}{101} a^{11} - \frac{317}{707} a^{10} + \frac{8}{101} a^{9} - \frac{166}{707} a^{8} - \frac{233}{707} a^{7} - \frac{38}{101} a^{6} - \frac{15}{101} a^{5} - \frac{106}{707} a^{4} + \frac{41}{707} a^{3} - \frac{300}{707} a^{2} + \frac{41}{707} a - \frac{327}{707}$, $\frac{1}{217049} a^{37} + \frac{85}{217049} a^{36} + \frac{6}{31007} a^{35} - \frac{15079}{217049} a^{34} + \frac{4682}{217049} a^{33} - \frac{8951}{217049} a^{32} + \frac{31237}{217049} a^{31} + \frac{56516}{217049} a^{30} + \frac{73692}{217049} a^{29} - \frac{83295}{217049} a^{28} - \frac{5202}{217049} a^{27} + \frac{75636}{217049} a^{26} + \frac{85413}{217049} a^{25} + \frac{80007}{217049} a^{24} - \frac{732}{2149} a^{23} + \frac{73543}{217049} a^{22} - \frac{22070}{217049} a^{21} + \frac{101443}{217049} a^{20} - \frac{42738}{217049} a^{19} + \frac{60680}{217049} a^{18} + \frac{18283}{217049} a^{17} + \frac{99079}{217049} a^{16} + \frac{97834}{217049} a^{15} - \frac{85977}{217049} a^{14} + \frac{96}{307} a^{13} - \frac{48942}{217049} a^{12} + \frac{23642}{217049} a^{11} + \frac{10713}{31007} a^{10} + \frac{62488}{217049} a^{9} - \frac{64233}{217049} a^{8} - \frac{51537}{217049} a^{7} + \frac{56629}{217049} a^{6} + \frac{146}{707} a^{5} - \frac{5850}{217049} a^{4} + \frac{78411}{217049} a^{3} + \frac{25367}{217049} a^{2} + \frac{70923}{217049} a - \frac{49330}{217049}$, $\frac{1}{217049} a^{38} - \frac{122}{217049} a^{36} + \frac{78}{217049} a^{35} - \frac{14669}{217049} a^{34} + \frac{5687}{217049} a^{33} - \frac{12882}{217049} a^{32} - \frac{37942}{217049} a^{31} - \frac{27235}{217049} a^{30} - \frac{72956}{217049} a^{29} + \frac{35363}{217049} a^{28} - \frac{78081}{217049} a^{27} - \frac{12079}{217049} a^{26} + \frac{4930}{217049} a^{25} + \frac{8526}{31007} a^{24} + \frac{63956}{217049} a^{23} - \frac{46037}{217049} a^{22} + \frac{47591}{217049} a^{21} - \frac{101935}{217049} a^{20} + \frac{29979}{217049} a^{19} + \frac{66589}{217049} a^{18} - \frac{67643}{217049} a^{17} + \frac{62745}{217049} a^{16} + \frac{95279}{217049} a^{15} + \frac{96947}{217049} a^{14} - \frac{67952}{217049} a^{13} - \frac{80211}{217049} a^{12} - \frac{19206}{217049} a^{11} - \frac{4054}{31007} a^{10} + \frac{65862}{217049} a^{9} + \frac{103308}{217049} a^{8} - \frac{3348}{31007} a^{7} - \frac{88428}{217049} a^{6} + \frac{49410}{217049} a^{5} + \frac{1277}{31007} a^{4} - \frac{98626}{217049} a^{3} - \frac{45564}{217049} a^{2} + \frac{100897}{217049} a - \frac{4868}{217049}$, $\frac{1}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{39} + \frac{9637707940341205688335695149829533744217394248374486335008621095518862309181895972683560734142059832303028796187033785467871810054184335789902983503841193166032671}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{38} + \frac{10908954458898538343709572349272468607140904506050931150627837021676735429793327611605156815684447477775033134945768041893243233017371184621213777290445412165315107}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{37} - \frac{5502396260426959415918202371965970256641210342613097433140336756067890202502774891551508771643283909823410328014795105587086807912540359759520320527518938939739908828}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{36} + \frac{765068059839002299543658304584856880713623647242149709145869211028897673399117300210387415615430222639568425866907720547755010659945704617051975439679321870010125412}{1194700663054419300578459783179072133343428700449006211017464705439680621772367141590476372655115865266627684098380166562642778633941554974996190423997231616774288799749} a^{35} - \frac{404375946982999868148622989511446892358467544115357454802938584821967480501362079658702480791193890767825771353764325581104168929589432136194746735575715628631087645118}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{34} - \frac{227897012530236696762120412969031211460611618831710863193008476442025943845182455323861866197935676599440024486859674668572680479254572555172024782321245531870124274273}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{33} - \frac{299139552288231095683056403165789135685859737291467631996085210286626806190072813882857237939422882728216648370644473170033772129964325926833492581480404049192696674999}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{32} - \frac{1570567497799451813486214656290876765144263763992613666889650886039525294965468988080518940730154536185046226133952047861745052343555461118254023695193821171888527890592}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{31} + \frac{3346876315093569077245221844149424402091855379978580006226214392612550858170320939741365857685928054951419541092779647929850235718531560392684888764697404166170977746}{27240731730882524768889962482910439522488602290368219795186491654976431115330846876655813057282772172203237096705736696868076385790198321905450595986907561294527757649} a^{30} + \frac{151705373238333627613589588744122426041310225698838618133388531624977165960111494146746738473156778024885265047573980289577866584307496758109098436469669886500508785961}{1194700663054419300578459783179072133343428700449006211017464705439680621772367141590476372655115865266627684098380166562642778633941554974996190423997231616774288799749} a^{29} + \frac{1052286065162112364509787298029530696382902728927954518175769270745005912800976509537646023009289396901048315560865002735502779636998288580166099351477908660400864275762}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{28} - \frac{2117764732479533594010645666451605439307626827381698707875368971667119772578535309875807739557619327077361311288382776854117949273944862576779728084188930933176421249190}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{27} + \frac{2248645367032313559908685996431041273161951530485106965124509433346721420704516326911228693695703090603357058785306808149557341047413850994771435444746154035292341770446}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{26} + \frac{1913707042444905510959270800106592556892983357443484249737193898711637451405048131544976499842772566053585366165176092377068938132246577648654699114905508053936498453225}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{25} - \frac{391249831141419432185376031934003699991722723719045168768666606341242700094260663352844092409191370624428668529277049527229173077659552769622013714889297284791618098304}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{24} + \frac{2795306252477355647058567605287216496328893550931253824630977642452818331293585577184106100601115565793287024834531574770015761994203113571584782212029166445557828770941}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{23} - \frac{2160780513891701514873956224627726882889198107345425654541654770134335881623430499981296136426757017568620013660909481300805082846500574412131732150731145566774734497081}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{22} + \frac{1353973995848421054034931115022402012629905639544365976498295923033618931185929239410077674012610099262946750094931163129684351577312605961608858922998104171931842041587}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{21} - \frac{1454266093303963032266945300238490478824536960264948359469122831510460146719038168509965182547686161308540058330465034828594522738919983535112964593754821960712574805875}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{20} + \frac{2371102147040978929347490987781001693974125133282610000527713489292364086474776312423506508385975633427264970523173596988836655783427774460461666593014085452042647907455}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{19} - \frac{189077348186545416452284328824068975479454527465088746551290289542165171397029060261677484714481944650576151357219307777024931857398730178539968000865599542875786817726}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{18} - \frac{2268002108697842848570258857122169919657350436761403082925159043200252823947785602348982486354739924526744858226195053331629007015544683030567406815640628872391227605282}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{17} - \frac{2284143332795338072974600981525475978737561612798840767853195421898777528968194658395149776156068322344476190241919499927503916465505327143267312029316761903856584894044}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{16} - \frac{3808236554903745354265648898121265476744572377147993447703293411164644837879646570257959367300738645028877570820082049825739072574374125125759392903735308865575658902102}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{15} + \frac{2438388213786983338372315322170260351155259113595305437496964642197519230558588019176754000510050316021690079237983159155829181236957032392444758551699588893827373251494}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{14} + \frac{2195878221435050339616279300694688012614538562949179199854170248705777268650117710193894094250845213593413098515119598459127342016754386628604882304801222476625474145180}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{13} + \frac{1424724405374989827416195929027520502024928325661303549741616176910542410344229915300725191818545075310266544683721418918807914520366597645032400786610564002936545326478}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{12} - \frac{1497622371181310279574877321833753619177421840957212009861871907989131092304632703409938953590984435538905645691685388514004980837201019108921396041978564753580703340443}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{11} + \frac{1158601908128258597611596682496139747212295491922770007092982403315430175643862848173880055305623522157099820593625054096047598977050832673628114276284847497652948244076}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{10} - \frac{74150434194543389964211785931123946646535068721908133114754938034479313563051163470090099048630872327491330085453915722695364050553278510434265395125761671329014589935}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{9} + \frac{489324062112136138354413002376815418377449332715971612898115729984728483950195378239034107261595628544928882032056835842162872275925207521079551924632598607695561224667}{1194700663054419300578459783179072133343428700449006211017464705439680621772367141590476372655115865266627684098380166562642778633941554974996190423997231616774288799749} a^{8} - \frac{3355760518425941929755213110973968623154380506350663287642977085819845994916787589099383319995655562414317414538589807097228142910149827173454558625986978292903928504093}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{7} + \frac{2610803978007193721868485097147928041724734445179093346182949891997792911601188621818186975567501166795653547901130395120868183264145238040424172751744915459240131184341}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{6} + \frac{2227881567048417116167113919696387089025606952058371656178035720120650572525748682397956656551173579074352400702437957818772456456171268090058770774450233980944162454180}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{5} + \frac{2341795326082793242966999065318404946726012469655082396230040848853533900922858653847638390458811134954958493884268176696618154160632999646843511037995857847665960323812}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{4} + \frac{3292016608614325251185533238857998616999404183556559158349353538481374870505485958553144419179254812180237325305233250268424142850459571140481108971915270756172524527400}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{3} - \frac{925257897921567253998275235629009287844677692540275368011243613866821324250152992040558188140558817008376701380414315269093617117212366218060104015146060824234003508829}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a^{2} - \frac{3735408100558071196801407858283047225671621753192920412370205320965217110411257144880579354119873560775378878444843830031512666235199335272459092638646072769640186114393}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243} a - \frac{522796993590534403121603515194924164573093951949796230986164340509175861538554199409587426794116311283963887371470328266213915215824834022775388914952235985596555882262}{8362904641380935104049218482253504933404000903143043477122252938077764352406569991133334608585811056866393788688661165938499450437590884824973332967980621317420021598243}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $39$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 21383108027423993000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{40}$ (as 40T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 40
The 40 conjugacy class representatives for $C_{40}$
Character table for $C_{40}$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 5.5.390625.1, \(\Q(\zeta_{17})^+\), 10.10.216652984619140625.1, 20.20.66645980149270403035916388034820556640625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20^{2}$ $40$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{5}$ $40$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ R $20^{2}$ $40$ $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
17Data not computed