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Properties

Label 40.40.6317295794...5161.1
Degree $40$
Signature $[40, 0]$
Discriminant $7^{20}\cdot 41^{39}$
Root discriminant $98.86$
Ramified primes $7, 41$
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![2308219, 40683397, -40683397, -1464023163, 1464023163, 14260160389, -14260160389, -63237601403, 63237601403, 156339390341, -156339390341, -242891503739, 242891503739, 256147113861, -256147113861, -192987641979, 192987641979, 107536349061, -107536349061, -45361821819, 45361821819, 14705316741, -14705316741, -3694696059, 3694696059, 721307013, -721307013, -109052539, 109052539, 12638085, -12638085, -1101179, 1101179, 69781, -69781, -3035, 3035, 81, -81, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 81*x^38 + 81*x^37 + 3035*x^36 - 3035*x^35 - 69781*x^34 + 69781*x^33 + 1101179*x^32 - 1101179*x^31 - 12638085*x^30 + 12638085*x^29 + 109052539*x^28 - 109052539*x^27 - 721307013*x^26 + 721307013*x^25 + 3694696059*x^24 - 3694696059*x^23 - 14705316741*x^22 + 14705316741*x^21 + 45361821819*x^20 - 45361821819*x^19 - 107536349061*x^18 + 107536349061*x^17 + 192987641979*x^16 - 192987641979*x^15 - 256147113861*x^14 + 256147113861*x^13 + 242891503739*x^12 - 242891503739*x^11 - 156339390341*x^10 + 156339390341*x^9 + 63237601403*x^8 - 63237601403*x^7 - 14260160389*x^6 + 14260160389*x^5 + 1464023163*x^4 - 1464023163*x^3 - 40683397*x^2 + 40683397*x + 2308219)
 
gp: K = bnfinit(x^40 - x^39 - 81*x^38 + 81*x^37 + 3035*x^36 - 3035*x^35 - 69781*x^34 + 69781*x^33 + 1101179*x^32 - 1101179*x^31 - 12638085*x^30 + 12638085*x^29 + 109052539*x^28 - 109052539*x^27 - 721307013*x^26 + 721307013*x^25 + 3694696059*x^24 - 3694696059*x^23 - 14705316741*x^22 + 14705316741*x^21 + 45361821819*x^20 - 45361821819*x^19 - 107536349061*x^18 + 107536349061*x^17 + 192987641979*x^16 - 192987641979*x^15 - 256147113861*x^14 + 256147113861*x^13 + 242891503739*x^12 - 242891503739*x^11 - 156339390341*x^10 + 156339390341*x^9 + 63237601403*x^8 - 63237601403*x^7 - 14260160389*x^6 + 14260160389*x^5 + 1464023163*x^4 - 1464023163*x^3 - 40683397*x^2 + 40683397*x + 2308219, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - 81 x^{38} + 81 x^{37} + 3035 x^{36} - 3035 x^{35} - 69781 x^{34} + 69781 x^{33} + 1101179 x^{32} - 1101179 x^{31} - 12638085 x^{30} + 12638085 x^{29} + 109052539 x^{28} - 109052539 x^{27} - 721307013 x^{26} + 721307013 x^{25} + 3694696059 x^{24} - 3694696059 x^{23} - 14705316741 x^{22} + 14705316741 x^{21} + 45361821819 x^{20} - 45361821819 x^{19} - 107536349061 x^{18} + 107536349061 x^{17} + 192987641979 x^{16} - 192987641979 x^{15} - 256147113861 x^{14} + 256147113861 x^{13} + 242891503739 x^{12} - 242891503739 x^{11} - 156339390341 x^{10} + 156339390341 x^{9} + 63237601403 x^{8} - 63237601403 x^{7} - 14260160389 x^{6} + 14260160389 x^{5} + 1464023163 x^{4} - 1464023163 x^{3} - 40683397 x^{2} + 40683397 x + 2308219 \)

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:  \(63172957949423116502957480067191906200305068755882825968063357506461803384975161=7^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.86$
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:  $7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(287=7\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{287}(1,·)$, $\chi_{287}(258,·)$, $\chi_{287}(6,·)$, $\chi_{287}(8,·)$, $\chi_{287}(265,·)$, $\chi_{287}(267,·)$, $\chi_{287}(13,·)$, $\chi_{287}(272,·)$, $\chi_{287}(148,·)$, $\chi_{287}(153,·)$, $\chi_{287}(155,·)$, $\chi_{287}(69,·)$, $\chi_{287}(34,·)$, $\chi_{287}(27,·)$, $\chi_{287}(36,·)$, $\chi_{287}(167,·)$, $\chi_{287}(169,·)$, $\chi_{287}(43,·)$, $\chi_{287}(48,·)$, $\chi_{287}(50,·)$, $\chi_{287}(181,·)$, $\chi_{287}(55,·)$, $\chi_{287}(57,·)$, $\chi_{287}(188,·)$, $\chi_{287}(64,·)$, $\chi_{287}(197,·)$, $\chi_{287}(97,·)$, $\chi_{287}(76,·)$, $\chi_{287}(202,·)$, $\chi_{287}(204,·)$, $\chi_{287}(162,·)$, $\chi_{287}(78,·)$, $\chi_{287}(141,·)$, $\chi_{287}(216,·)$, $\chi_{287}(92,·)$, $\chi_{287}(225,·)$, $\chi_{287}(104,·)$, $\chi_{287}(111,·)$, $\chi_{287}(113,·)$, $\chi_{287}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{703889} a^{21} - \frac{159413}{703889} a^{20} - \frac{42}{703889} a^{19} + \frac{41519}{703889} a^{18} + \frac{756}{703889} a^{17} - \frac{1934}{703889} a^{16} - \frac{7616}{703889} a^{15} + \frac{308039}{703889} a^{14} + \frac{47040}{703889} a^{13} + \frac{227716}{703889} a^{12} - \frac{183456}{703889} a^{11} - \frac{182138}{703889} a^{10} - \frac{255441}{703889} a^{9} + \frac{88629}{703889} a^{8} + \frac{44945}{703889} a^{7} + \frac{215790}{703889} a^{6} - \frac{171665}{703889} a^{5} - \frac{310841}{703889} a^{4} - \frac{197120}{703889} a^{3} - \frac{351244}{703889} a^{2} + \frac{21504}{703889} a + \frac{126672}{703889}$, $\frac{1}{703889} a^{22} - \frac{44}{703889} a^{20} - \frac{318826}{703889} a^{19} + \frac{836}{703889} a^{18} + \frac{149275}{703889} a^{17} - \frac{8976}{703889} a^{16} - \frac{276733}{703889} a^{15} + \frac{59840}{703889} a^{14} - \frac{218170}{703889} a^{13} - \frac{256256}{703889} a^{12} - \frac{273294}{703889} a^{11} + \frac{815}{703889} a^{10} + \frac{155035}{703889} a^{9} + \frac{199714}{703889} a^{8} + \frac{146944}{703889} a^{7} - \frac{199714}{703889} a^{6} - \frac{146944}{703889} a^{5} + \frac{84369}{703889} a^{4} - \frac{125177}{703889} a^{3} + \frac{123904}{703889} a^{2} + \frac{204394}{703889} a - \frac{4096}{703889}$, $\frac{1}{703889} a^{23} - \frac{294108}{703889} a^{20} - \frac{1012}{703889} a^{19} - \frac{135556}{703889} a^{18} + \frac{24288}{703889} a^{17} + \frac{342060}{703889} a^{16} - \frac{275264}{703889} a^{15} - \frac{38345}{703889} a^{14} - \frac{298163}{703889} a^{13} - \frac{108236}{703889} a^{12} - \frac{328470}{703889} a^{11} - \frac{116258}{703889} a^{10} + \frac{222534}{703889} a^{9} - \frac{176714}{703889} a^{8} - \frac{333801}{703889} a^{7} + \frac{197259}{703889} a^{6} + \frac{273888}{703889} a^{5} + \frac{275599}{703889} a^{4} - \frac{102708}{703889} a^{3} + \frac{235216}{703889} a^{2} + \frac{238191}{703889} a - \frac{57544}{703889}$, $\frac{1}{703889} a^{24} - \frac{1104}{703889} a^{20} + \frac{181910}{703889} a^{19} + \frac{27968}{703889} a^{18} + \frac{258784}{703889} a^{17} - \frac{337824}{703889} a^{16} - \frac{190075}{703889} a^{15} + \frac{290637}{703889} a^{14} - \frac{206211}{703889} a^{13} - \frac{157825}{703889} a^{12} - \frac{86100}{703889} a^{11} + \frac{44197}{703889} a^{10} + \frac{62406}{703889} a^{9} - \frac{253317}{703889} a^{8} - \frac{154101}{703889} a^{7} - \frac{312477}{703889} a^{6} + \frac{72082}{703889} a^{5} + \frac{175784}{703889} a^{4} + \frac{75963}{703889} a^{3} + \frac{21368}{703889} a^{2} - \frac{1777}{703889} a - \frac{188416}{703889}$, $\frac{1}{703889} a^{25} + \frac{162208}{703889} a^{20} - \frac{18400}{703889} a^{19} + \frac{342975}{703889} a^{18} - \frac{207089}{703889} a^{17} - \frac{213544}{703889} a^{16} + \frac{329241}{703889} a^{15} - \frac{109542}{703889} a^{14} - \frac{313451}{703889} a^{13} + \frac{23991}{703889} a^{12} + \frac{228805}{703889} a^{11} + \frac{294308}{703889} a^{10} - \frac{692}{703889} a^{9} - \frac{148256}{703889} a^{8} + \frac{34573}{703889} a^{7} - \frac{314129}{703889} a^{6} + \frac{3765}{703889} a^{5} - \frac{298558}{703889} a^{4} - \frac{97411}{703889} a^{3} + \frac{67686}{703889} a^{2} + \frac{323663}{703889} a - \frac{228023}{703889}$, $\frac{1}{703889} a^{26} - \frac{20800}{703889} a^{20} + \frac{116821}{703889} a^{19} - \frac{111089}{703889} a^{18} + \frac{337783}{703889} a^{17} + \frac{105019}{703889} a^{16} - \frac{58609}{703889} a^{15} + \frac{264880}{703889} a^{14} - \frac{83569}{703889} a^{13} + \frac{151041}{703889} a^{12} + \frac{9903}{703889} a^{11} - \frac{92985}{703889} a^{10} - \frac{513}{703889} a^{9} - \frac{69323}{703889} a^{8} + \frac{129573}{703889} a^{7} + \frac{131637}{703889} a^{6} - \frac{7189}{703889} a^{5} - \frac{177331}{703889} a^{4} + \frac{350821}{703889} a^{3} + \frac{23088}{703889} a^{2} + \frac{125029}{703889} a + \frac{12023}{703889}$, $\frac{1}{703889} a^{27} + \frac{347500}{703889} a^{20} - \frac{280800}{703889} a^{19} + \frac{261180}{703889} a^{18} + \frac{344261}{703889} a^{17} - \frac{164136}{703889} a^{16} + \frac{227105}{703889} a^{15} + \frac{329953}{703889} a^{14} + \frac{177331}{703889} a^{13} + \frac{33622}{703889} a^{12} - \frac{195516}{703889} a^{11} - \frac{140315}{703889} a^{10} - \frac{287951}{703889} a^{9} + \frac{127482}{703889} a^{8} + \frac{223045}{703889} a^{7} - \frac{275342}{703889} a^{6} + \frac{19566}{703889} a^{5} + \frac{78486}{703889} a^{4} + \frac{80513}{703889} a^{3} - \frac{86240}{703889} a^{2} + \frac{325708}{703889} a + \frac{121073}{703889}$, $\frac{1}{703889} a^{28} - \frac{327600}{703889} a^{20} + \frac{74511}{703889} a^{19} + \frac{104594}{703889} a^{18} - \frac{323539}{703889} a^{17} + \frac{78110}{703889} a^{16} + \frac{267313}{703889} a^{15} - \frac{159383}{703889} a^{14} + \frac{47869}{703889} a^{13} - \frac{304136}{703889} a^{12} + \frac{296844}{703889} a^{11} - \frac{327942}{703889} a^{10} - \frac{159030}{703889} a^{9} + \frac{308740}{703889} a^{8} - \frac{69821}{703889} a^{7} - \frac{302486}{703889} a^{6} - \frac{222875}{703889} a^{5} - \frac{70149}{703889} a^{4} + \frac{155725}{703889} a^{3} - \frac{256337}{703889} a^{2} - \frac{33303}{703889} a - \frac{117496}{703889}$, $\frac{1}{703889} a^{29} + \frac{12288}{703889} a^{20} - \frac{280715}{703889} a^{19} + \frac{53714}{703889} a^{18} - \frac{25218}{703889} a^{17} + \frac{189013}{703889} a^{16} + \frac{125522}{703889} a^{15} - \frac{126105}{703889} a^{14} - \frac{242013}{703889} a^{13} - \frac{209443}{703889} a^{12} + \frac{344834}{703889} a^{11} + \frac{102700}{703889} a^{10} - \frac{319095}{703889} a^{9} + \frac{73218}{703889} a^{8} - \frac{270588}{703889} a^{7} + \frac{304966}{703889} a^{6} - \frac{312494}{703889} a^{5} + \frac{265755}{703889} a^{4} + \frac{120190}{703889} a^{3} - \frac{17317}{703889} a^{2} + \frac{71792}{703889} a - \frac{28795}{703889}$, $\frac{1}{703889} a^{30} - \frac{336858}{703889} a^{20} - \frac{134079}{703889} a^{19} + \frac{108835}{703889} a^{18} + \frac{49842}{703889} a^{17} - \frac{41712}{703889} a^{16} - \frac{157934}{703889} a^{15} + \frac{89797}{703889} a^{14} - \frac{344094}{703889} a^{13} + \frac{129401}{703889} a^{12} - \frac{146439}{703889} a^{11} + \frac{129518}{703889} a^{10} + \frac{291175}{703889} a^{9} + \frac{276432}{703889} a^{8} - \frac{130218}{703889} a^{7} + \frac{313738}{703889} a^{6} + \frac{129942}{703889} a^{5} - \frac{271205}{703889} a^{4} + \frac{111194}{703889} a^{3} - \frac{89284}{703889} a^{2} - \frac{311572}{703889} a - \frac{246957}{703889}$, $\frac{1}{703889} a^{31} + \frac{13377}{703889} a^{20} + \frac{38579}{703889} a^{19} - \frac{217286}{703889} a^{18} - \frac{184882}{703889} a^{17} + \frac{159908}{703889} a^{16} + \frac{254674}{703889} a^{15} - \frac{147345}{703889} a^{14} - \frac{19447}{703889} a^{13} + \frac{98336}{703889} a^{12} + \frac{146914}{703889} a^{11} + \frac{133456}{703889} a^{10} - \frac{157141}{703889} a^{9} - \frac{194471}{703889} a^{8} - \frac{255842}{703889} a^{7} + \frac{100732}{703889} a^{6} + \frac{297131}{703889} a^{5} - \frac{46522}{703889} a^{4} - \frac{169429}{703889} a^{3} - \frac{145358}{703889} a^{2} - \frac{174224}{703889} a + \frac{21507}{703889}$, $\frac{1}{703889} a^{32} - \frac{277390}{703889} a^{20} + \frac{344548}{703889} a^{19} - \frac{216124}{703889} a^{18} - \frac{98658}{703889} a^{17} + \frac{81899}{703889} a^{16} - \frac{332018}{703889} a^{15} - \frac{90944}{703889} a^{14} + \frac{121022}{703889} a^{13} - \frac{282315}{703889} a^{12} - \frac{236575}{703889} a^{11} + \frac{143056}{703889} a^{10} + \frac{162580}{703889} a^{9} + \frac{206990}{703889} a^{8} - \frac{7327}{703889} a^{7} + \frac{323090}{703889} a^{6} + \frac{230265}{703889} a^{5} + \frac{78305}{703889} a^{4} - \frac{39312}{703889} a^{3} - \frac{42311}{703889} a^{2} + \frac{253100}{703889} a - \frac{230521}{703889}$, $\frac{1}{703889} a^{33} - \frac{216653}{703889} a^{20} + \frac{99609}{703889} a^{19} - \frac{175066}{703889} a^{18} + \frac{29817}{703889} a^{17} + \frac{263029}{703889} a^{16} - \frac{322295}{703889} a^{15} - \frac{138145}{703889} a^{14} + \frac{152892}{703889} a^{13} + \frac{313583}{703889} a^{12} + \frac{346249}{703889} a^{11} - \frac{56487}{703889} a^{10} - \frac{289704}{703889} a^{9} + \frac{59880}{703889} a^{8} + \frac{334672}{703889} a^{7} + \frac{201694}{703889} a^{6} + \frac{14805}{703889} a^{5} + \frac{66531}{703889} a^{4} + \frac{346187}{703889} a^{3} + \frac{291431}{703889} a^{2} + \frac{8653}{703889} a + \frac{111089}{703889}$, $\frac{1}{703889} a^{34} - \frac{187406}{703889} a^{20} - \frac{123935}{703889} a^{19} + \frac{248193}{703889} a^{18} + \frac{46560}{703889} a^{17} + \frac{188647}{703889} a^{16} - \frac{251577}{703889} a^{15} - \frac{101398}{703889} a^{14} + \frac{61872}{703889} a^{13} + \frac{120787}{703889} a^{12} + \frac{150908}{703889} a^{11} - \frac{312589}{703889} a^{10} - \frac{134246}{703889} a^{9} - \frac{18511}{703889} a^{8} + \frac{70353}{703889} a^{7} - \frac{37816}{703889} a^{6} - \frac{287621}{703889} a^{5} + \frac{291089}{703889} a^{4} + \frac{5479}{703889} a^{3} + \frac{86000}{703889} a^{2} - \frac{24090}{703889} a - \frac{59405}{703889}$, $\frac{1}{703889} a^{35} + \frac{84214}{703889} a^{20} + \frac{119920}{703889} a^{19} + \frac{167268}{703889} a^{18} - \frac{317995}{703889} a^{17} - \frac{191946}{703889} a^{16} + \frac{101398}{703889} a^{15} - \frac{333740}{703889} a^{14} + \frac{193191}{703889} a^{13} + \frac{113312}{703889} a^{12} - \frac{313409}{703889} a^{11} - \frac{198997}{703889} a^{10} + \frac{296333}{703889} a^{9} + \frac{7994}{703889} a^{8} + \frac{189080}{703889} a^{7} + \frac{222291}{703889} a^{6} - \frac{217045}{703889} a^{5} - \frac{313216}{703889} a^{4} + \frac{117778}{703889} a^{3} + \frac{330459}{703889} a^{2} + \frac{154694}{703889} a - \frac{267582}{703889}$, $\frac{1}{703889} a^{36} - \frac{348595}{703889} a^{20} + \frac{184811}{703889} a^{19} + \frac{121491}{703889} a^{18} + \frac{196169}{703889} a^{17} - \frac{330974}{703889} a^{16} - \frac{202795}{703889} a^{15} + \frac{122051}{703889} a^{14} + \frac{174044}{703889} a^{13} + \frac{267172}{703889} a^{12} - \frac{295074}{703889} a^{11} - \frac{283223}{703889} a^{10} + \frac{164639}{703889} a^{9} - \frac{278459}{703889} a^{8} + \frac{35214}{703889} a^{7} + \frac{250097}{703889} a^{6} - \frac{189188}{703889} a^{5} - \frac{350158}{703889} a^{4} + \frac{75963}{703889} a^{3} + \frac{289463}{703889} a^{2} - \frac{99041}{703889} a - \frac{118013}{703889}$, $\frac{1}{703889} a^{37} + \frac{238848}{703889} a^{20} + \frac{262170}{703889} a^{19} + \frac{146356}{703889} a^{18} - \frac{47640}{703889} a^{17} - \frac{59863}{703889} a^{16} + \frac{291839}{703889} a^{15} - \frac{53257}{703889} a^{14} - \frac{326061}{703889} a^{13} - \frac{14140}{703889} a^{12} - \frac{292448}{703889} a^{11} - \frac{35893}{703889} a^{10} - \frac{255909}{703889} a^{9} - \frac{138408}{703889} a^{8} - \frac{12879}{703889} a^{7} - \frac{83790}{703889} a^{6} - \frac{83609}{703889} a^{5} - \frac{165883}{703889} a^{4} + \frac{295021}{703889} a^{3} + \frac{194218}{703889} a^{2} - \frac{348983}{703889} a + \frac{157203}{703889}$, $\frac{1}{703889} a^{38} + \frac{270717}{703889} a^{20} + \frac{323526}{703889} a^{19} + \frac{314369}{703889} a^{18} + \frac{270522}{703889} a^{17} - \frac{231202}{703889} a^{16} + \frac{163935}{703889} a^{15} - \frac{123519}{703889} a^{14} + \frac{52158}{703889} a^{13} - \frac{300586}{703889} a^{12} + \frac{268656}{703889} a^{11} - \frac{114641}{703889} a^{10} - \frac{257182}{703889} a^{9} - \frac{114485}{703889} a^{8} - \frac{96011}{703889} a^{7} - \frac{229282}{703889} a^{6} + \frac{141787}{703889} a^{5} - \frac{53864}{703889} a^{4} + \frac{184546}{703889} a^{3} - \frac{136425}{703889} a^{2} + \frac{247844}{703889} a - \frac{92969}{703889}$, $\frac{1}{703889} a^{39} - \frac{5832}{703889} a^{20} - \frac{281630}{703889} a^{19} + \frac{70951}{703889} a^{18} - \frac{61555}{703889} a^{17} + \frac{37197}{703889} a^{16} - \frac{33728}{703889} a^{15} - \frac{204197}{703889} a^{14} - \frac{68478}{703889} a^{13} + \frac{274904}{703889} a^{12} + \frac{247138}{703889} a^{11} + \frac{171314}{703889} a^{10} - \frac{60315}{703889} a^{9} - \frac{8661}{703889} a^{8} - \frac{179593}{703889} a^{7} - \frac{19866}{703889} a^{6} - \frac{283506}{703889} a^{5} + \frac{197593}{703889} a^{4} - \frac{338142}{703889} a^{3} + \frac{308671}{703889} a^{2} + \frac{274582}{703889} a - \frac{199522}{703889}$

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:  $39$
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:  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, 8.8.467605011588281.1, 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 $20^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{5}$ $20^{2}$ R $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ $40$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$ R $20^{2}$ $40$ $40$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
7Data not computed
41Data not computed