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Properties

Label 40.0.22027048434...3561.1
Degree $40$
Signature $[0, 20]$
Discriminant $3^{20}\cdot 7^{20}\cdot 41^{39}$
Root discriminant $171.23$
Ramified primes $3, 7, 41$
Class number Not computed
Class group Not computed
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![1896660547630982431, -1892750482933716806, 1892750482933716806, -1838009577171998056, 1838009577171998056, -1609192591088013681, 1609192591088013681, -1158096247093873056, 1158096247093873056, -646853723900513681, 646853723900513681, -275040979759888681, 275040979759888681, -89134607689576181, 89134607689576181, -22208313744263681, 22208313744263681, -4295688011841806, 4295688011841806, -650311546998056, 650311546998056, -77466673951181, 77466673951181, -7276195435556, 7276195435556, -537909498056, 537909498056, -31098248056, 31098248056, -1388623056, 1388623056, -46898056, 46898056, -1157431, 1157431, -19681, 19681, -206, 206, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 206*x^38 - 206*x^37 + 19681*x^36 - 19681*x^35 + 1157431*x^34 - 1157431*x^33 + 46898056*x^32 - 46898056*x^31 + 1388623056*x^30 - 1388623056*x^29 + 31098248056*x^28 - 31098248056*x^27 + 537909498056*x^26 - 537909498056*x^25 + 7276195435556*x^24 - 7276195435556*x^23 + 77466673951181*x^22 - 77466673951181*x^21 + 650311546998056*x^20 - 650311546998056*x^19 + 4295688011841806*x^18 - 4295688011841806*x^17 + 22208313744263681*x^16 - 22208313744263681*x^15 + 89134607689576181*x^14 - 89134607689576181*x^13 + 275040979759888681*x^12 - 275040979759888681*x^11 + 646853723900513681*x^10 - 646853723900513681*x^9 + 1158096247093873056*x^8 - 1158096247093873056*x^7 + 1609192591088013681*x^6 - 1609192591088013681*x^5 + 1838009577171998056*x^4 - 1838009577171998056*x^3 + 1892750482933716806*x^2 - 1892750482933716806*x + 1896660547630982431)
 
gp: K = bnfinit(x^40 - x^39 + 206*x^38 - 206*x^37 + 19681*x^36 - 19681*x^35 + 1157431*x^34 - 1157431*x^33 + 46898056*x^32 - 46898056*x^31 + 1388623056*x^30 - 1388623056*x^29 + 31098248056*x^28 - 31098248056*x^27 + 537909498056*x^26 - 537909498056*x^25 + 7276195435556*x^24 - 7276195435556*x^23 + 77466673951181*x^22 - 77466673951181*x^21 + 650311546998056*x^20 - 650311546998056*x^19 + 4295688011841806*x^18 - 4295688011841806*x^17 + 22208313744263681*x^16 - 22208313744263681*x^15 + 89134607689576181*x^14 - 89134607689576181*x^13 + 275040979759888681*x^12 - 275040979759888681*x^11 + 646853723900513681*x^10 - 646853723900513681*x^9 + 1158096247093873056*x^8 - 1158096247093873056*x^7 + 1609192591088013681*x^6 - 1609192591088013681*x^5 + 1838009577171998056*x^4 - 1838009577171998056*x^3 + 1892750482933716806*x^2 - 1892750482933716806*x + 1896660547630982431, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 206 x^{38} - 206 x^{37} + 19681 x^{36} - 19681 x^{35} + 1157431 x^{34} - 1157431 x^{33} + 46898056 x^{32} - 46898056 x^{31} + 1388623056 x^{30} - 1388623056 x^{29} + 31098248056 x^{28} - 31098248056 x^{27} + 537909498056 x^{26} - 537909498056 x^{25} + 7276195435556 x^{24} - 7276195435556 x^{23} + 77466673951181 x^{22} - 77466673951181 x^{21} + 650311546998056 x^{20} - 650311546998056 x^{19} + 4295688011841806 x^{18} - 4295688011841806 x^{17} + 22208313744263681 x^{16} - 22208313744263681 x^{15} + 89134607689576181 x^{14} - 89134607689576181 x^{13} + 275040979759888681 x^{12} - 275040979759888681 x^{11} + 646853723900513681 x^{10} - 646853723900513681 x^{9} + 1158096247093873056 x^{8} - 1158096247093873056 x^{7} + 1609192591088013681 x^{6} - 1609192591088013681 x^{5} + 1838009577171998056 x^{4} - 1838009577171998056 x^{3} + 1892750482933716806 x^{2} - 1892750482933716806 x + 1896660547630982431 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(220270484343077469571317811864553170412678895179244714569241039133217272745060389147263561=3^{20}\cdot 7^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $171.23$
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:  $3, 7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(861=3\cdot 7\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{861}(1,·)$, $\chi_{861}(776,·)$, $\chi_{861}(778,·)$, $\chi_{861}(631,·)$, $\chi_{861}(652,·)$, $\chi_{861}(398,·)$, $\chi_{861}(272,·)$, $\chi_{861}(148,·)$, $\chi_{861}(799,·)$, $\chi_{861}(545,·)$, $\chi_{861}(293,·)$, $\chi_{861}(295,·)$, $\chi_{861}(169,·)$, $\chi_{861}(43,·)$, $\chi_{861}(440,·)$, $\chi_{861}(314,·)$, $\chi_{861}(671,·)$, $\chi_{861}(188,·)$, $\chi_{861}(650,·)$, $\chi_{861}(64,·)$, $\chi_{861}(608,·)$, $\chi_{861}(587,·)$, $\chi_{861}(839,·)$, $\chi_{861}(841,·)$, $\chi_{861}(715,·)$, $\chi_{861}(335,·)$, $\chi_{861}(337,·)$, $\chi_{861}(356,·)$, $\chi_{861}(442,·)$, $\chi_{861}(736,·)$, $\chi_{861}(400,·)$, $\chi_{861}(610,·)$, $\chi_{861}(484,·)$, $\chi_{861}(104,·)$, $\chi_{861}(167,·)$, $\chi_{861}(755,·)$, $\chi_{861}(629,·)$, $\chi_{861}(503,·)$, $\chi_{861}(379,·)$, $\chi_{861}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{413885273661951521} a^{21} + \frac{81373671544827399}{413885273661951521} a^{20} + \frac{105}{413885273661951521} a^{19} - \frac{140338318756290520}{413885273661951521} a^{18} + \frac{4725}{413885273661951521} a^{17} - \frac{169984715875025806}{413885273661951521} a^{16} + \frac{119000}{413885273661951521} a^{15} - \frac{31210984888954381}{413885273661951521} a^{14} + \frac{1837500}{413885273661951521} a^{13} - \frac{94565491737548509}{413885273661951521} a^{12} + \frac{17915625}{413885273661951521} a^{11} + \frac{78371274765866467}{413885273661951521} a^{10} + \frac{109484375}{413885273661951521} a^{9} - \frac{200981795675785494}{413885273661951521} a^{8} + \frac{402187500}{413885273661951521} a^{7} - \frac{13496279034949297}{413885273661951521} a^{6} + \frac{812109375}{413885273661951521} a^{5} - \frac{202162743219212067}{413885273661951521} a^{4} + \frac{751953125}{413885273661951521} a^{3} + \frac{140858663106567897}{413885273661951521} a^{2} + \frac{205078125}{413885273661951521} a + \frac{138251448409242246}{413885273661951521}$, $\frac{1}{413885273661951521} a^{22} + \frac{110}{413885273661951521} a^{20} + \frac{7016915937814526}{413885273661951521} a^{19} + \frac{5225}{413885273661951521} a^{18} - \frac{161163533231523072}{413885273661951521} a^{17} + \frac{140250}{413885273661951521} a^{16} + \frac{175623049330301456}{413885273661951521} a^{15} + \frac{2337500}{413885273661951521} a^{14} + \frac{116786741142780661}{413885273661951521} a^{13} + \frac{25025000}{413885273661951521} a^{12} - \frac{137313459791657491}{413885273661951521} a^{11} + \frac{172046875}{413885273661951521} a^{10} + \frac{44511415862273842}{413885273661951521} a^{9} + \frac{737343750}{413885273661951521} a^{8} - \frac{49261325966090896}{413885273661951521} a^{7} + \frac{1843359375}{413885273661951521} a^{6} - \frac{123153314915227240}{413885273661951521} a^{5} + \frac{2363281250}{413885273661951521} a^{4} + \frac{198686456956111199}{413885273661951521} a^{3} + \frac{1181640625}{413885273661951521} a^{2} - \frac{71732938901946774}{413885273661951521} a + \frac{97656250}{413885273661951521}$, $\frac{1}{413885273661951521} a^{23} + \frac{161389066569734098}{413885273661951521} a^{20} - \frac{6325}{413885273661951521} a^{19} - \frac{37703595531772149}{413885273661951521} a^{18} - \frac{379500}{413885273661951521} a^{17} - \frac{164780792866629850}{413885273661951521} a^{16} - \frac{10752500}{413885273661951521} a^{15} - \frac{174972384029801118}{413885273661951521} a^{14} - \frac{177100000}{413885273661951521} a^{13} - \frac{82241210210109526}{413885273661951521} a^{12} - \frac{1798671875}{413885273661951521} a^{11} + \frac{115261938517944413}{413885273661951521} a^{10} - \frac{11305937500}{413885273661951521} a^{9} + \frac{122816694286882831}{413885273661951521} a^{8} - \frac{42397265625}{413885273661951521} a^{7} + \frac{119781557943340867}{413885273661951521} a^{6} - \frac{86968750000}{413885273661951521} a^{5} + \frac{86783433324056435}{413885273661951521} a^{4} - \frac{81533203125}{413885273661951521} a^{3} + \frac{161454518529742354}{413885273661951521} a^{2} - \frac{22460937500}{413885273661951521} a + \frac{106095800475559217}{413885273661951521}$, $\frac{1}{413885273661951521} a^{24} - \frac{6900}{413885273661951521} a^{20} - \frac{14259365213840078}{413885273661951521} a^{19} - \frac{437000}{413885273661951521} a^{18} + \frac{62439024116410303}{413885273661951521} a^{17} - \frac{13196250}{413885273661951521} a^{16} + \frac{44459553148965845}{413885273661951521} a^{15} - \frac{234600000}{413885273661951521} a^{14} + \frac{31483154607175663}{413885273661951521} a^{13} - \frac{2616250000}{413885273661951521} a^{12} + \frac{85889992916108323}{413885273661951521} a^{11} - \frac{18500625000}{413885273661951521} a^{10} + \frac{35295976422762619}{413885273661951521} a^{9} - \frac{80940234375}{413885273661951521} a^{8} + \frac{93909218405109542}{413885273661951521} a^{7} - \frac{205562500000}{413885273661951521} a^{6} + \frac{1279219264201624}{4986569562192187} a^{5} - \frac{266835937500}{413885273661951521} a^{4} + \frac{133422604423486960}{413885273661951521} a^{3} - \frac{134765625000}{413885273661951521} a^{2} - \frac{164930804792843222}{413885273661951521} a - \frac{11230468750}{413885273661951521}$, $\frac{1}{413885273661951521} a^{25} - \frac{178242065173000975}{413885273661951521} a^{20} + \frac{287500}{413885273661951521} a^{19} - \frac{194305298983570078}{413885273661951521} a^{18} + \frac{19406250}{413885273661951521} a^{17} + \frac{100785573441514959}{413885273661951521} a^{16} + \frac{586500000}{413885273661951521} a^{15} - \frac{103970274963262317}{413885273661951521} a^{14} + \frac{10062500000}{413885273661951521} a^{13} - \frac{132811704933006681}{413885273661951521} a^{12} + \frac{105117187500}{413885273661951521} a^{11} - \frac{150960815269253028}{413885273661951521} a^{10} + \frac{674501953125}{413885273661951521} a^{9} - \frac{164814176977203708}{413885273661951521} a^{8} + \frac{2569531250000}{413885273661951521} a^{7} + \frac{106036431717677717}{413885273661951521} a^{6} + \frac{5336718750000}{413885273661951521} a^{5} + \frac{3866632636850430}{413885273661951521} a^{4} + \frac{5053710937500}{413885273661951521} a^{3} - \frac{42777927736525230}{413885273661951521} a^{2} + \frac{1403808593750}{413885273661951521} a - \frac{70561767026758505}{413885273661951521}$, $\frac{1}{413885273661951521} a^{26} + \frac{325000}{413885273661951521} a^{20} - \frac{103725770606286148}{413885273661951521} a^{19} + \frac{23156250}{413885273661951521} a^{18} + \frac{38011613799776599}{413885273661951521} a^{17} + \frac{745875000}{413885273661951521} a^{16} - \frac{90719315538785525}{413885273661951521} a^{15} + \frac{13812500000}{413885273661951521} a^{14} - \frac{171663227738557111}{413885273661951521} a^{13} + \frac{158437500000}{413885273661951521} a^{12} + \frac{91064747869557561}{413885273661951521} a^{11} + \frac{1143720703125}{413885273661951521} a^{10} + \frac{72575110107450884}{413885273661951521} a^{9} + \frac{5083203125000}{413885273661951521} a^{8} + \frac{65841781100270825}{413885273661951521} a^{7} + \frac{13071093750000}{413885273661951521} a^{6} - \frac{109132971391525142}{413885273661951521} a^{5} + \frac{17138671875000}{413885273661951521} a^{4} - \frac{86090853886954238}{413885273661951521} a^{3} + \frac{8728027343750}{413885273661951521} a^{2} - \frac{120000317218871100}{413885273661951521} a + \frac{732421875000}{413885273661951521}$, $\frac{1}{413885273661951521} a^{27} - \frac{105761388132672290}{413885273661951521} a^{20} - \frac{10968750}{413885273661951521} a^{19} - \frac{165550138838837601}{413885273661951521} a^{18} - \frac{789750000}{413885273661951521} a^{17} - \frac{50503055778907084}{413885273661951521} a^{16} - \frac{24862500000}{413885273661951521} a^{15} - \frac{101861224672608779}{413885273661951521} a^{14} - \frac{438750000000}{413885273661951521} a^{13} - \frac{2886864399112336}{413885273661951521} a^{12} - \frac{4678857421875}{413885273661951521} a^{11} - \frac{91982639997721776}{413885273661951521} a^{10} - \frac{30499218750000}{413885273661951521} a^{9} + \frac{189432355858728126}{413885273661951521} a^{8} - \frac{117639843750000}{413885273661951521} a^{7} - \frac{174576882232219700}{413885273661951521} a^{6} - \frac{246796875000000}{413885273661951521} a^{5} + \frac{173802649878668096}{413885273661951521} a^{4} - \frac{235656738281250}{413885273661951521} a^{3} - \frac{163160750651561332}{413885273661951521} a^{2} - \frac{65917968750000}{413885273661951521} a + \frac{78461011389121281}{413885273661951521}$, $\frac{1}{413885273661951521} a^{28} - \frac{12796875}{413885273661951521} a^{20} + \frac{178378499881013303}{413885273661951521} a^{19} - \frac{972562500}{413885273661951521} a^{18} + \frac{112530561122177319}{413885273661951521} a^{17} - \frac{32632031250}{413885273661951521} a^{16} + \frac{79925050708050653}{413885273661951521} a^{15} - \frac{621562500000}{413885273661951521} a^{14} + \frac{28641140892689282}{413885273661951521} a^{13} - \frac{7278222656250}{413885273661951521} a^{12} + \frac{8472774766368239}{413885273661951521} a^{11} - \frac{53373632812500}{413885273661951521} a^{10} + \frac{205492273016459354}{413885273661951521} a^{9} - \frac{240181347656250}{413885273661951521} a^{8} + \frac{129371240718560201}{413885273661951521} a^{7} - \frac{623847656250000}{413885273661951521} a^{6} - \frac{111860485005344063}{413885273661951521} a^{5} - \frac{824798583984375}{413885273661951521} a^{4} - \frac{121081969127904797}{413885273661951521} a^{3} - \frac{422973632812500}{413885273661951521} a^{2} - \frac{98192627132586719}{413885273661951521} a - \frac{35705566406250}{413885273661951521}$, $\frac{1}{413885273661951521} a^{29} + \frac{155059621566979764}{413885273661951521} a^{20} + \frac{371109375}{413885273661951521} a^{19} - \frac{149930901971418976}{413885273661951521} a^{18} + \frac{27833203125}{413885273661951521} a^{17} - \frac{106727378728076578}{413885273661951521} a^{16} + \frac{901265625000}{413885273661951521} a^{15} - \frac{29558451026341404}{413885273661951521} a^{14} + \frac{16236035156250}{413885273661951521} a^{13} - \frac{172356932735550076}{413885273661951521} a^{12} + \frac{175890380859375}{413885273661951521} a^{11} - \frac{144153332014567255}{413885273661951521} a^{10} + \frac{1160876513671875}{413885273661951521} a^{9} + \frac{177879984339978286}{413885273661951521} a^{8} + \frac{4522895507812500}{413885273661951521} a^{7} - \frac{121789456040192848}{413885273661951521} a^{6} + \frac{9567663574218750}{413885273661951521} a^{5} + \frac{96214480064991707}{413885273661951521} a^{4} + \frac{9199676513671875}{413885273661951521} a^{3} - \frac{54291655368214918}{413885273661951521} a^{2} + \frac{2588653564453125}{413885273661951521} a - \frac{36998430189757972}{413885273661951521}$, $\frac{1}{413885273661951521} a^{30} + \frac{445331250}{413885273661951521} a^{20} + \frac{124219779973766644}{413885273661951521} a^{19} + \frac{35255390625}{413885273661951521} a^{18} - \frac{186504901053269308}{413885273661951521} a^{17} + \frac{1216708593750}{413885273661951521} a^{16} + \frac{122630749166403339}{413885273661951521} a^{15} + \frac{23658222656250}{413885273661951521} a^{14} + \frac{120370092332720013}{413885273661951521} a^{13} + \frac{281424609375000}{413885273661951521} a^{12} - \frac{83790434817106654}{413885273661951521} a^{11} + \frac{2089577724609375}{413885273661951521} a^{10} - \frac{143044433334348354}{413885273661951521} a^{9} + \frac{9498080566406250}{413885273661951521} a^{8} + \frac{202806126213859714}{413885273661951521} a^{7} + \frac{24875925292968750}{413885273661951521} a^{6} - \frac{60204015643537834}{413885273661951521} a^{5} + \frac{33118835449218750}{413885273661951521} a^{4} - \frac{107149090580864155}{413885273661951521} a^{3} + \frac{17085113525390625}{413885273661951521} a^{2} + \frac{99089641538410971}{413885273661951521} a + \frac{1449645996093750}{413885273661951521}$, $\frac{1}{413885273661951521} a^{31} + \frac{25169143245840455}{413885273661951521} a^{20} - \frac{11504390625}{413885273661951521} a^{19} - \frac{158322005959051704}{413885273661951521} a^{18} - \frac{887481562500}{413885273661951521} a^{17} + \frac{78747969684796946}{413885273661951521} a^{16} - \frac{29336196093750}{413885273661951521} a^{15} + \frac{159066577537620585}{413885273661951521} a^{14} - \frac{536871562500000}{413885273661951521} a^{13} + \frac{804281320377893}{413885273661951521} a^{12} - \frac{5888809951171875}{413885273661951521} a^{11} - \frac{60827369183606774}{413885273661951521} a^{10} - \frac{39258733007812500}{413885273661951521} a^{9} + \frac{170242942836316440}{413885273661951521} a^{8} - \frac{154230736816406250}{413885273661951521} a^{7} - \frac{128561097292751637}{413885273661951521} a^{6} + \frac{85346426005701521}{413885273661951521} a^{5} - \frac{136353541846144812}{413885273661951521} a^{4} + \frac{96102162089685896}{413885273661951521} a^{3} + \frac{144099864053688076}{413885273661951521} a^{2} - \frac{89878051757812500}{413885273661951521} a - \frac{202441614579448050}{413885273661951521}$, $\frac{1}{413885273661951521} a^{32} - \frac{14159250000}{413885273661951521} a^{20} + \frac{96114868861361168}{413885273661951521} a^{19} - \frac{1152967500000}{413885273661951521} a^{18} - \frac{60380325931266402}{413885273661951521} a^{17} - \frac{40619348437500}{413885273661951521} a^{16} - \frac{95139459595318459}{413885273661951521} a^{15} - \frac{802357500000000}{413885273661951521} a^{14} + \frac{68339583271174975}{413885273661951521} a^{13} - \frac{9663688125000000}{413885273661951521} a^{12} - \frac{23186085656262506}{413885273661951521} a^{11} - \frac{72477660937500000}{413885273661951521} a^{10} + \frac{124349493062119286}{413885273661951521} a^{9} + \frac{81695994365076521}{413885273661951521} a^{8} + \frac{40716249383219243}{413885273661951521} a^{7} - \frac{48333046426096958}{413885273661951521} a^{6} - \frac{176591459823005016}{413885273661951521} a^{5} + \frac{68302793642104563}{413885273661951521} a^{4} + \frac{60855279964526607}{413885273661951521} a^{3} - \frac{194519999775548479}{413885273661951521} a^{2} + \frac{151114656725282023}{413885273661951521} a - \frac{51852722167968750}{413885273661951521}$, $\frac{1}{413885273661951521} a^{33} + \frac{114746505075534903}{413885273661951521} a^{20} + \frac{333753750000}{413885273661951521} a^{19} + \frac{147242200838301014}{413885273661951521} a^{18} + \frac{26283107812500}{413885273661951521} a^{17} + \frac{37227181210314780}{413885273661951521} a^{16} + \frac{882593250000000}{413885273661951521} a^{15} - \frac{79541025461072567}{413885273661951521} a^{14} + \frac{16353933750000000}{413885273661951521} a^{13} - \frac{134040611569765549}{413885273661951521} a^{12} + \frac{181194152343750000}{413885273661951521} a^{11} + \frac{142154321811976158}{413885273661951521} a^{10} - \frac{23628463563979563}{413885273661951521} a^{9} + \frac{195225728209159846}{413885273661951521} a^{8} - \frac{148053518318418252}{413885273661951521} a^{7} - \frac{146900281225289414}{413885273661951521} a^{6} - \frac{21625200923788025}{413885273661951521} a^{5} + \frac{55512087315713295}{413885273661951521} a^{4} + \frac{105440443831913496}{413885273661951521} a^{3} + \frac{110250424180975332}{413885273661951521} a^{2} - \frac{45297196395379397}{413885273661951521} a + \frac{37302104111764483}{413885273661951521}$, $\frac{1}{413885273661951521} a^{34} + \frac{420282500000}{413885273661951521} a^{20} + \frac{101532104103730308}{413885273661951521} a^{19} + \frac{34935982812500}{413885273661951521} a^{18} + \frac{49699196464390615}{413885273661951521} a^{17} + \frac{1250340437500000}{413885273661951521} a^{16} - \frac{10696359009948735}{413885273661951521} a^{15} + \frac{25006808750000000}{413885273661951521} a^{14} - \frac{20499478004830456}{413885273661951521} a^{13} - \frac{109658054911951521}{413885273661951521} a^{12} + \frac{110995871499726632}{413885273661951521} a^{11} - \frac{182593300174834126}{413885273661951521} a^{10} - \frac{179136034215454600}{413885273661951521} a^{9} - \frac{142317076148239546}{413885273661951521} a^{8} + \frac{99647786257651234}{413885273661951521} a^{7} + \frac{27862719112296572}{413885273661951521} a^{6} + \frac{171301586352926661}{413885273661951521} a^{5} - \frac{153516465962039932}{413885273661951521} a^{4} - \frac{150435097743339570}{413885273661951521} a^{3} - \frac{114446932160391758}{413885273661951521} a^{2} + \frac{41457920616397059}{413885273661951521} a + \frac{34473553789693916}{413885273661951521}$, $\frac{1}{413885273661951521} a^{35} + \frac{93775246856700019}{413885273661951521} a^{20} - \frac{9193679687500}{413885273661951521} a^{19} + \frac{101784019085264757}{413885273661951521} a^{18} - \frac{735494375000000}{413885273661951521} a^{17} + \frac{133482234067444695}{413885273661951521} a^{16} - \frac{25006808750000000}{413885273661951521} a^{15} - \frac{85221338536580102}{413885273661951521} a^{14} - \frac{54156601338048479}{413885273661951521} a^{13} + \frac{44370293991686365}{413885273661951521} a^{12} + \frac{151603235339744773}{413885273661951521} a^{11} - \frac{103224678427086516}{413885273661951521} a^{10} + \frac{198466738052830806}{413885273661951521} a^{9} - \frac{84948912964593346}{413885273661951521} a^{8} - \frac{139313595561482860}{413885273661951521} a^{7} - \frac{46477641232328452}{413885273661951521} a^{6} - \frac{13524093289535107}{413885273661951521} a^{5} - \frac{50962037579477193}{413885273661951521} a^{4} + \frac{61162887758070286}{413885273661951521} a^{3} + \frac{176996491684547261}{413885273661951521} a^{2} - \frac{68136594836889716}{413885273661951521} a - \frac{69223197625575284}{413885273661951521}$, $\frac{1}{413885273661951521} a^{36} - \frac{11820445312500}{413885273661951521} a^{20} + \frac{188629667018599266}{413885273661951521} a^{19} - \frac{998170937500000}{413885273661951521} a^{18} - \frac{97316345552017610}{413885273661951521} a^{17} - \frac{36170562656250000}{413885273661951521} a^{16} - \frac{164848812301931900}{413885273661951521} a^{15} + \frac{97052109823903042}{413885273661951521} a^{14} + \frac{56484240659608753}{413885273661951521} a^{13} + \frac{141638325250433462}{413885273661951521} a^{12} + \frac{14886301585156767}{413885273661951521} a^{11} + \frac{27998474534500965}{413885273661951521} a^{10} - \frac{190930137991454103}{413885273661951521} a^{9} + \frac{100429683648615086}{413885273661951521} a^{8} + \frac{96341781948651335}{413885273661951521} a^{7} - \frac{8111026044994118}{413885273661951521} a^{6} - \frac{136665276581570344}{413885273661951521} a^{5} - \frac{59718342691694269}{413885273661951521} a^{4} - \frac{9661948711230125}{413885273661951521} a^{3} + \frac{109441613170496324}{413885273661951521} a^{2} - \frac{195381912944778484}{413885273661951521} a + \frac{17757820800738604}{413885273661951521}$, $\frac{1}{413885273661951521} a^{37} - \frac{58829655778369317}{413885273661951521} a^{20} + \frac{242975820312500}{413885273661951521} a^{19} - \frac{10132205531784650}{413885273661951521} a^{18} + \frac{19681041445312500}{413885273661951521} a^{17} - \frac{184490324399594789}{413885273661951521} a^{16} - \frac{151855992636403042}{413885273661951521} a^{15} - \frac{89168481779202178}{413885273661951521} a^{14} - \frac{74212917114247151}{413885273661951521} a^{13} + \frac{77982596727751372}{413885273661951521} a^{12} - \frac{110596088626865287}{413885273661951521} a^{11} - \frac{49568014398573100}{413885273661951521} a^{10} + \frac{35246203468396419}{413885273661951521} a^{9} + \frac{114558026799886135}{413885273661951521} a^{8} + \frac{140984813873585676}{413885273661951521} a^{7} + \frac{142246008778664475}{413885273661951521} a^{6} + \frac{193584571721366678}{413885273661951521} a^{5} + \frac{141590809021545109}{413885273661951521} a^{4} - \frac{69903924923806172}{413885273661951521} a^{3} - \frac{178965442863451346}{413885273661951521} a^{2} + \frac{6471335289742607}{413885273661951521} a + \frac{84418701740523018}{413885273661951521}$, $\frac{1}{413885273661951521} a^{38} + \frac{318382109375000}{413885273661951521} a^{20} - \frac{41297453732279180}{413885273661951521} a^{19} + \frac{27221670351562500}{413885273661951521} a^{18} + \frac{826681942481415}{4986569562192187} a^{17} + \frac{168620735879221958}{413885273661951521} a^{16} + \frac{184350425921494628}{413885273661951521} a^{15} + \frac{16481063220625471}{413885273661951521} a^{14} + \frac{186929774525581050}{413885273661951521} a^{13} - \frac{88564706330121726}{413885273661951521} a^{12} - \frac{22046215024612542}{413885273661951521} a^{11} - \frac{101985851249821461}{413885273661951521} a^{10} + \frac{52810537606338700}{413885273661951521} a^{9} - \frac{62048698837215297}{413885273661951521} a^{8} + \frac{202604197547395361}{413885273661951521} a^{7} + \frac{173775317194062061}{413885273661951521} a^{6} - \frac{102661359588705409}{413885273661951521} a^{5} - \frac{74988055518847666}{413885273661951521} a^{4} + \frac{70076752257446217}{413885273661951521} a^{3} + \frac{103725867938069343}{413885273661951521} a^{2} + \frac{152339300409858717}{413885273661951521} a + \frac{175279562729613282}{413885273661951521}$, $\frac{1}{413885273661951521} a^{39} - \frac{40993874952559273}{413885273661951521} a^{20} - \frac{6208451132812500}{413885273661951521} a^{19} + \frac{98972479197948145}{413885273661951521} a^{18} - \frac{94078909931798479}{413885273661951521} a^{17} - \frac{180980854916706706}{413885273661951521} a^{16} + \frac{206455224495165403}{413885273661951521} a^{15} - \frac{82702504468131504}{413885273661951521} a^{14} + \frac{118086275106828968}{413885273661951521} a^{13} - \frac{53309230197283320}{413885273661951521} a^{12} + \frac{50377486281665961}{413885273661951521} a^{11} - \frac{56749886448383538}{413885273661951521} a^{10} - \frac{96671719133790156}{413885273661951521} a^{9} - \frac{29567839606943896}{413885273661951521} a^{8} - \frac{63217585071016396}{413885273661951521} a^{7} + \frac{42571017541996190}{413885273661951521} a^{6} - \frac{4337495041128109}{413885273661951521} a^{5} + \frac{115927534297063675}{413885273661951521} a^{4} - \frac{106780462209046896}{413885273661951521} a^{3} + \frac{92333807232849278}{413885273661951521} a^{2} - \frac{145512792368114842}{413885273661951521} a - \frac{47389076683896096}{413885273661951521}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{41}) \), 4.4.68921.1, 5.5.2825761.1, 8.0.37876005938650761.1, 10.10.327381934393961.1, \(\Q(\zeta_{41})^+\)

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 $20^{2}$ R $20^{2}$ R $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{8}$ $40$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$ R $20^{2}$ $40$ $40$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
7Data not computed
41Data not computed