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Properties

Label 40.0.18497731891...8961.1
Degree $40$
Signature $[0, 20]$
Discriminant $3^{20}\cdot 11^{36}\cdot 23^{20}$
Root discriminant $71.89$
Ramified primes $3, 11, 23$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3656158440062976, -609359740010496, 609359740010496, -287753210560512, 47958868426752, -47958868426752, 5720583979008, -953430663168, 953430663168, 881959643136, -146993273856, -148978579968, -46569033216, -33345696384, 7311135168, -417991104, 3304954224, 114302232, 248494824, -105733914, -42843857, -17622319, 6902634, 529177, 2550119, -53754, 156703, -119119, -27726, -14783, -2431, 2431, 438, -73, 73, -102, 17, -17, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 6*x^38 - 17*x^37 + 17*x^36 - 102*x^35 + 73*x^34 - 73*x^33 + 438*x^32 + 2431*x^31 - 2431*x^30 - 14783*x^29 - 27726*x^28 - 119119*x^27 + 156703*x^26 - 53754*x^25 + 2550119*x^24 + 529177*x^23 + 6902634*x^22 - 17622319*x^21 - 42843857*x^20 - 105733914*x^19 + 248494824*x^18 + 114302232*x^17 + 3304954224*x^16 - 417991104*x^15 + 7311135168*x^14 - 33345696384*x^13 - 46569033216*x^12 - 148978579968*x^11 - 146993273856*x^10 + 881959643136*x^9 + 953430663168*x^8 - 953430663168*x^7 + 5720583979008*x^6 - 47958868426752*x^5 + 47958868426752*x^4 - 287753210560512*x^3 + 609359740010496*x^2 - 609359740010496*x + 3656158440062976)
 
gp: K = bnfinit(x^40 - x^39 + 6*x^38 - 17*x^37 + 17*x^36 - 102*x^35 + 73*x^34 - 73*x^33 + 438*x^32 + 2431*x^31 - 2431*x^30 - 14783*x^29 - 27726*x^28 - 119119*x^27 + 156703*x^26 - 53754*x^25 + 2550119*x^24 + 529177*x^23 + 6902634*x^22 - 17622319*x^21 - 42843857*x^20 - 105733914*x^19 + 248494824*x^18 + 114302232*x^17 + 3304954224*x^16 - 417991104*x^15 + 7311135168*x^14 - 33345696384*x^13 - 46569033216*x^12 - 148978579968*x^11 - 146993273856*x^10 + 881959643136*x^9 + 953430663168*x^8 - 953430663168*x^7 + 5720583979008*x^6 - 47958868426752*x^5 + 47958868426752*x^4 - 287753210560512*x^3 + 609359740010496*x^2 - 609359740010496*x + 3656158440062976, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 6 x^{38} - 17 x^{37} + 17 x^{36} - 102 x^{35} + 73 x^{34} - 73 x^{33} + 438 x^{32} + 2431 x^{31} - 2431 x^{30} - 14783 x^{29} - 27726 x^{28} - 119119 x^{27} + 156703 x^{26} - 53754 x^{25} + 2550119 x^{24} + 529177 x^{23} + 6902634 x^{22} - 17622319 x^{21} - 42843857 x^{20} - 105733914 x^{19} + 248494824 x^{18} + 114302232 x^{17} + 3304954224 x^{16} - 417991104 x^{15} + 7311135168 x^{14} - 33345696384 x^{13} - 46569033216 x^{12} - 148978579968 x^{11} - 146993273856 x^{10} + 881959643136 x^{9} + 953430663168 x^{8} - 953430663168 x^{7} + 5720583979008 x^{6} - 47958868426752 x^{5} + 47958868426752 x^{4} - 287753210560512 x^{3} + 609359740010496 x^{2} - 609359740010496 x + 3656158440062976 \)

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:  \(184977318917298023500939014698578314068923764241165004603173974059593268961=3^{20}\cdot 11^{36}\cdot 23^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.89$
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:  $3, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(759=3\cdot 11\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{759}(1,·)$, $\chi_{759}(643,·)$, $\chi_{759}(392,·)$, $\chi_{759}(137,·)$, $\chi_{759}(139,·)$, $\chi_{759}(530,·)$, $\chi_{759}(277,·)$, $\chi_{759}(668,·)$, $\chi_{759}(413,·)$, $\chi_{759}(415,·)$, $\chi_{759}(160,·)$, $\chi_{759}(551,·)$, $\chi_{759}(553,·)$, $\chi_{759}(298,·)$, $\chi_{759}(47,·)$, $\chi_{759}(689,·)$, $\chi_{759}(691,·)$, $\chi_{759}(436,·)$, $\chi_{759}(185,·)$, $\chi_{759}(574,·)$, $\chi_{759}(323,·)$, $\chi_{759}(68,·)$, $\chi_{759}(70,·)$, $\chi_{759}(712,·)$, $\chi_{759}(461,·)$, $\chi_{759}(206,·)$, $\chi_{759}(208,·)$, $\chi_{759}(599,·)$, $\chi_{759}(344,·)$, $\chi_{759}(346,·)$, $\chi_{759}(91,·)$, $\chi_{759}(482,·)$, $\chi_{759}(229,·)$, $\chi_{759}(620,·)$, $\chi_{759}(622,·)$, $\chi_{759}(367,·)$, $\chi_{759}(116,·)$, $\chi_{759}(758,·)$, $\chi_{759}(505,·)$, $\chi_{759}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{6} a^{21} - \frac{1}{6} a^{20} + \frac{1}{6} a^{18} - \frac{1}{6} a^{17} + \frac{1}{6} a^{15} - \frac{1}{6} a^{14} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{182124} a^{22} - \frac{1}{36} a^{21} - \frac{1}{3} a^{20} + \frac{1}{36} a^{19} + \frac{17}{36} a^{18} - \frac{1}{3} a^{17} - \frac{17}{36} a^{16} - \frac{1}{36} a^{15} - \frac{1}{3} a^{14} + \frac{1}{36} a^{13} + \frac{17}{36} a^{12} + \frac{78907}{182124} a^{11} + \frac{1}{3} a^{10} - \frac{13}{36} a^{9} - \frac{5}{36} a^{8} + \frac{1}{3} a^{7} + \frac{5}{36} a^{6} + \frac{13}{36} a^{5} + \frac{1}{3} a^{4} - \frac{13}{36} a^{3} - \frac{5}{36} a^{2} + \frac{1}{3} a + \frac{168}{5059}$, $\frac{1}{1092744} a^{23} - \frac{1}{1092744} a^{22} + \frac{1}{36} a^{21} + \frac{1}{216} a^{20} + \frac{107}{216} a^{19} - \frac{17}{36} a^{18} - \frac{17}{216} a^{17} - \frac{91}{216} a^{16} + \frac{1}{36} a^{15} + \frac{73}{216} a^{14} + \frac{35}{216} a^{13} + \frac{352093}{1092744} a^{12} + \frac{83813}{182124} a^{11} + \frac{95}{216} a^{10} + \frac{13}{216} a^{9} + \frac{5}{36} a^{8} - \frac{103}{216} a^{7} - \frac{5}{216} a^{6} - \frac{13}{36} a^{5} + \frac{23}{216} a^{4} + \frac{85}{216} a^{3} + \frac{5}{36} a^{2} - \frac{5003}{10118} a - \frac{28}{5059}$, $\frac{1}{6556464} a^{24} - \frac{1}{6556464} a^{23} + \frac{1}{1092744} a^{22} - \frac{107}{1296} a^{21} + \frac{107}{1296} a^{20} - \frac{107}{216} a^{19} + \frac{307}{1296} a^{18} - \frac{307}{1296} a^{17} + \frac{91}{216} a^{16} - \frac{251}{1296} a^{15} + \frac{251}{1296} a^{14} + \frac{3083953}{6556464} a^{13} + \frac{539123}{1092744} a^{12} - \frac{2172415}{6556464} a^{11} - \frac{635}{1296} a^{10} - \frac{13}{216} a^{9} - \frac{211}{1296} a^{8} + \frac{211}{1296} a^{7} + \frac{5}{216} a^{6} - \frac{85}{1296} a^{5} + \frac{85}{1296} a^{4} - \frac{85}{216} a^{3} - \frac{1677}{10118} a^{2} + \frac{1677}{10118} a + \frac{28}{5059}$, $\frac{1}{39338784} a^{25} - \frac{1}{39338784} a^{24} + \frac{1}{6556464} a^{23} - \frac{17}{39338784} a^{22} + \frac{539}{7776} a^{21} - \frac{107}{1296} a^{20} + \frac{2467}{7776} a^{19} + \frac{2717}{7776} a^{18} + \frac{307}{1296} a^{17} - \frac{3275}{7776} a^{16} + \frac{683}{7776} a^{15} - \frac{3472511}{39338784} a^{14} - \frac{3103357}{6556464} a^{13} - \frac{13099855}{39338784} a^{12} + \frac{13137439}{39338784} a^{11} + \frac{635}{1296} a^{10} + \frac{1517}{7776} a^{9} + \frac{1075}{7776} a^{8} - \frac{211}{1296} a^{7} + \frac{347}{7776} a^{6} + \frac{2245}{7776} a^{5} - \frac{85}{1296} a^{4} + \frac{65795}{182124} a^{3} - \frac{5087}{182124} a^{2} - \frac{1677}{10118} a + \frac{1111}{5059}$, $\frac{1}{236032704} a^{26} - \frac{1}{236032704} a^{25} + \frac{1}{39338784} a^{24} - \frac{17}{236032704} a^{23} + \frac{17}{236032704} a^{22} - \frac{539}{7776} a^{21} + \frac{10243}{46656} a^{20} + \frac{20861}{46656} a^{19} - \frac{2717}{7776} a^{18} - \frac{11051}{46656} a^{17} - \frac{4501}{46656} a^{16} + \frac{101430913}{236032704} a^{15} + \frac{3453107}{39338784} a^{14} + \frac{39351857}{236032704} a^{13} + \frac{39363295}{236032704} a^{12} - \frac{12989695}{39338784} a^{11} - \frac{21811}{46656} a^{10} - \frac{9293}{46656} a^{9} - \frac{1075}{7776} a^{8} + \frac{347}{46656} a^{7} + \frac{15205}{46656} a^{6} - \frac{2245}{7776} a^{5} - \frac{116329}{1092744} a^{4} + \frac{480577}{1092744} a^{3} + \frac{5087}{182124} a^{2} + \frac{3085}{15177} a + \frac{658}{5059}$, $\frac{1}{1416196224} a^{27} - \frac{1}{1416196224} a^{26} + \frac{1}{236032704} a^{25} - \frac{17}{1416196224} a^{24} + \frac{17}{1416196224} a^{23} - \frac{17}{236032704} a^{22} - \frac{20861}{279936} a^{21} + \frac{20861}{279936} a^{20} - \frac{20861}{46656} a^{19} - \frac{88811}{279936} a^{18} + \frac{88811}{279936} a^{17} - \frac{449312063}{1416196224} a^{16} - \frac{22772749}{236032704} a^{15} + \frac{39351857}{1416196224} a^{14} - \frac{39314273}{1416196224} a^{13} + \frac{39462017}{236032704} a^{12} + \frac{870503}{1416196224} a^{11} - \frac{55949}{279936} a^{10} + \frac{9293}{46656} a^{9} - \frac{61861}{279936} a^{8} + \frac{61861}{279936} a^{7} - \frac{15205}{46656} a^{6} + \frac{2797655}{6556464} a^{5} - \frac{2797655}{6556464} a^{4} - \frac{480577}{1092744} a^{3} + \frac{14519}{30354} a^{2} - \frac{14519}{30354} a - \frac{658}{5059}$, $\frac{1}{8497177344} a^{28} - \frac{1}{8497177344} a^{27} + \frac{1}{1416196224} a^{26} - \frac{17}{8497177344} a^{25} + \frac{17}{8497177344} a^{24} - \frac{17}{1416196224} a^{23} + \frac{73}{8497177344} a^{22} + \frac{20861}{1679616} a^{21} + \frac{25795}{279936} a^{20} + \frac{750997}{1679616} a^{19} + \frac{648683}{1679616} a^{18} - \frac{3281704511}{8497177344} a^{17} + \frac{449292659}{1416196224} a^{16} + \frac{2871744305}{8497177344} a^{15} - \frac{2871706721}{8497177344} a^{14} + \frac{39462017}{1416196224} a^{13} - \frac{1415325721}{8497177344} a^{12} + \frac{1418405017}{8497177344} a^{11} + \frac{9293}{279936} a^{10} - \frac{341797}{1679616} a^{9} + \frac{621733}{1679616} a^{8} - \frac{108517}{279936} a^{7} + \frac{9354119}{39338784} a^{6} - \frac{2797655}{39338784} a^{5} + \frac{1704911}{6556464} a^{4} - \frac{15835}{182124} a^{3} + \frac{46189}{182124} a^{2} + \frac{4730}{15177} a + \frac{591}{5059}$, $\frac{1}{50983064064} a^{29} - \frac{1}{50983064064} a^{28} + \frac{1}{8497177344} a^{27} - \frac{17}{50983064064} a^{26} + \frac{17}{50983064064} a^{25} - \frac{17}{8497177344} a^{24} + \frac{73}{50983064064} a^{23} - \frac{73}{50983064064} a^{22} - \frac{20861}{1679616} a^{21} - \frac{928619}{10077696} a^{20} - \frac{750997}{10077696} a^{19} + \frac{12296453953}{50983064064} a^{18} + \frac{3281685107}{8497177344} a^{17} - \frac{21203591503}{50983064064} a^{16} - \frac{21282257633}{50983064064} a^{15} + \frac{2871854465}{8497177344} a^{14} + \frac{8498047847}{50983064064} a^{13} + \frac{2208793}{50983064064} a^{12} - \frac{1416725401}{8497177344} a^{11} - \frac{341797}{10077696} a^{10} + \frac{2021413}{10077696} a^{9} - \frac{621733}{1679616} a^{8} + \frac{88031687}{236032704} a^{7} - \frac{48692903}{236032704} a^{6} + \frac{2797655}{39338784} a^{5} - \frac{15835}{1092744} a^{4} + \frac{197959}{1092744} a^{3} - \frac{46189}{182124} a^{2} - \frac{9527}{30354} a + \frac{2431}{5059}$, $\frac{1}{1529491921920} a^{30} + \frac{1}{305898384384} a^{29} + \frac{19}{1529491921920} a^{27} - \frac{17}{305898384384} a^{26} - \frac{539}{1529491921920} a^{24} + \frac{73}{305898384384} a^{23} + \frac{1}{302330880} a^{21} + \frac{10828693}{60466176} a^{20} + \frac{6397315547}{305898384384} a^{19} - \frac{539}{7080981120} a^{18} + \frac{1816723}{60466176} a^{17} + \frac{121548995393}{305898384384} a^{16} + \frac{19}{32782320} a^{15} - \frac{11701115}{60466176} a^{14} - \frac{83270851609}{305898384384} a^{13} + \frac{1}{151770} a^{12} - \frac{12094685}{60466176} a^{11} - \frac{12099109}{60466176} a^{10} + \frac{1}{5} a^{9} - \frac{1703594945}{8497177344} a^{8} + \frac{284725607}{1416196224} a^{7} - \frac{1}{5} a^{6} + \frac{8189737}{39338784} a^{5} - \frac{1290703}{6556464} a^{4} + \frac{1}{5} a^{3} - \frac{42509}{182124} a^{2} - \frac{2431}{30354} a - \frac{1}{5}$, $\frac{1}{917219979778649425920} a^{31} - \frac{125315581}{917219979778649425920} a^{30} + \frac{88014187}{10191333108651660288} a^{29} - \frac{8547854741}{917219979778649425920} a^{28} + \frac{48906132341}{917219979778649425920} a^{27} - \frac{1496241179}{10191333108651660288} a^{26} + \frac{145313530381}{917219979778649425920} a^{25} - \frac{804336084301}{917219979778649425920} a^{24} + \frac{6425035651}{10191333108651660288} a^{23} - \frac{623993392421}{917219979778649425920} a^{22} - \frac{10655840768752201}{181304601656186880} a^{21} + \frac{52975286902638781781}{183443995955729885184} a^{20} + \frac{19105026323357223457}{50956665543258301440} a^{19} - \frac{279560632658767954411}{917219979778649425920} a^{18} - \frac{80244240097193562145}{183443995955729885184} a^{17} + \frac{13644626038647730423}{50956665543258301440} a^{16} - \frac{113322815807279312029}{917219979778649425920} a^{15} + \frac{10909888247457376889}{183443995955729885184} a^{14} + \frac{23628908218042239937}{50956665543258301440} a^{13} - \frac{59374101930534505291}{917219979778649425920} a^{12} + \frac{86996136904154694095}{183443995955729885184} a^{11} - \frac{3360077511920723}{10072477869788160} a^{10} + \frac{261321235252883}{943641954504783360} a^{9} + \frac{23480693850942139}{70773146587858752} a^{8} - \frac{77982724738962623}{235910488626195840} a^{7} - \frac{13071201798763}{4368712752336960} a^{6} + \frac{114539454959701}{327653456425272} a^{5} - \frac{354998423490617}{1092178188084240} a^{4} - \frac{181068540157}{20225522001560} a^{3} + \frac{199236135670}{505638050039} a^{2} - \frac{1111424224769}{15169141501170} a - \frac{509519377516}{2528190250195}$, $\frac{1}{5503319878671896555520} a^{32} - \frac{1}{5503319878671896555520} a^{31} - \frac{262300697}{917219979778649425920} a^{30} + \frac{31110785299}{5503319878671896555520} a^{29} - \frac{38979806239}{5503319878671896555520} a^{28} + \frac{33996092977}{917219979778649425920} a^{27} - \frac{528883350299}{5503319878671896555520} a^{26} + \frac{662656706279}{5503319878671896555520} a^{25} - \frac{521276630057}{917219979778649425920} a^{24} + \frac{2271087330499}{5503319878671896555520} a^{23} - \frac{2845525859119}{5503319878671896555520} a^{22} - \frac{454600669871118566003}{5503319878671896555520} a^{21} + \frac{186181294844070706541}{917219979778649425920} a^{20} + \frac{917031828877774896029}{5503319878671896555520} a^{19} - \frac{2456410802814563674577}{5503319878671896555520} a^{18} + \frac{32451302026911904799}{917219979778649425920} a^{17} + \frac{2342406179570650320011}{5503319878671896555520} a^{16} + \frac{2369731373211778836457}{5503319878671896555520} a^{15} - \frac{409152710516200253959}{917219979778649425920} a^{14} - \frac{1257025307408881095091}{5503319878671896555520} a^{13} + \frac{503830861546720047583}{5503319878671896555520} a^{12} - \frac{53885159037567119281}{917219979778649425920} a^{11} - \frac{272713354098242983}{629094636336522240} a^{10} - \frac{10185970578349879673}{25478332771629150720} a^{9} + \frac{246592679130096809}{1061597198817881280} a^{8} - \frac{674291542989217}{2912475168224640} a^{7} + \frac{46913865731782273}{117955244313097920} a^{6} + \frac{2179210541961371}{4914801846379080} a^{5} - \frac{51927339873127}{121353132009360} a^{4} - \frac{222151315454873}{546089094042120} a^{3} + \frac{13141192097653}{45507424503510} a^{2} + \frac{2141258763287}{7584570750585} a + \frac{17338242028}{2528190250195}$, $\frac{1}{33019919272031379333120} a^{33} - \frac{1}{33019919272031379333120} a^{32} + \frac{1}{5503319878671896555520} a^{31} + \frac{9942566167}{33019919272031379333120} a^{30} + \frac{323332495001}{33019919272031379333120} a^{29} - \frac{45603277361}{5503319878671896555520} a^{28} + \frac{1830626741953}{33019919272031379333120} a^{27} - \frac{5496652414801}{33019919272031379333120} a^{26} + \frac{775255714921}{5503319878671896555520} a^{25} - \frac{33268248905273}{33019919272031379333120} a^{24} + \frac{23603272131401}{33019919272031379333120} a^{23} - \frac{70062604573287551}{33019919272031379333120} a^{22} - \frac{430768031405901201337}{5503319878671896555520} a^{21} + \frac{12257222039010750993449}{33019919272031379333120} a^{20} + \frac{6485412358923277561591}{33019919272031379333120} a^{19} - \frac{2149563257408036513983}{5503319878671896555520} a^{18} - \frac{1212834655387998246289}{33019919272031379333120} a^{17} - \frac{15127822272277728976991}{33019919272031379333120} a^{16} - \frac{2491206928514237105977}{5503319878671896555520} a^{15} + \frac{14651770477784102251529}{33019919272031379333120} a^{14} + \frac{12021083622391715964391}{33019919272031379333120} a^{13} + \frac{350752547063434714097}{5503319878671896555520} a^{12} + \frac{720555157300738133}{76434998314887452160} a^{11} - \frac{15298120038301147069}{38217499157443726080} a^{10} + \frac{5951218047916868123}{25478332771629150720} a^{9} - \frac{998055098510730601}{4246388795271525120} a^{8} + \frac{71329384086205349}{176932866469646880} a^{7} + \frac{50800129074398957}{117955244313097920} a^{6} - \frac{8131075353763519}{19659207385516320} a^{5} - \frac{319948188593029}{819133641063180} a^{4} + \frac{123073740639023}{546089094042120} a^{3} - \frac{18290040903871}{91014849007020} a^{2} - \frac{3329385139532}{7584570750585} a + \frac{1219021251933}{2528190250195}$, $\frac{1}{198119515632188275998720} a^{34} - \frac{1}{198119515632188275998720} a^{33} + \frac{1}{33019919272031379333120} a^{32} - \frac{17}{198119515632188275998720} a^{31} + \frac{905824429}{39623903126437655199744} a^{30} - \frac{161819130041}{33019919272031379333120} a^{29} + \frac{993560390857}{198119515632188275998720} a^{28} - \frac{1175061804869}{39623903126437655199744} a^{27} + \frac{2750925210481}{33019919272031379333120} a^{26} - \frac{16890526640897}{198119515632188275998720} a^{25} + \frac{19780392606109}{39623903126437655199744} a^{24} - \frac{70113507116761391}{198119515632188275998720} a^{23} + \frac{11685860041049267}{33019919272031379333120} a^{22} + \frac{2621145275294757593581}{39623903126437655199744} a^{21} - \frac{70751119385888455890929}{198119515632188275998720} a^{20} - \frac{13756991955653746060087}{33019919272031379333120} a^{19} - \frac{14689713379062683245445}{39623903126437655199744} a^{18} + \frac{9678406252351954923049}{198119515632188275998720} a^{17} + \frac{14969510724692107699007}{33019919272031379333120} a^{16} + \frac{548972991899216557021}{39623903126437655199744} a^{15} + \frac{60625693015614262846111}{198119515632188275998720} a^{14} + \frac{9395199794870212779833}{33019919272031379333120} a^{13} + \frac{20486505029390011531}{183443995955729885184} a^{12} - \frac{21848206430514400001}{917219979778649425920} a^{11} - \frac{4349765792277650011}{19108749578721863040} a^{10} - \frac{269317640645519051}{2547833277162915072} a^{9} - \frac{68911068760427827}{2123194397635762560} a^{8} - \frac{29136521772166729}{88466433234823440} a^{7} - \frac{46539042521141}{11795524431309792} a^{6} + \frac{3512042831714867}{9829603692758160} a^{5} - \frac{190436717692637}{819133641063180} a^{4} - \frac{2750596184599}{27304454702106} a^{3} - \frac{1442738591396}{22753712251755} a^{2} - \frac{137178932798}{7584570750585} a - \frac{821763089861}{2528190250195}$, $\frac{1}{1188717093793129655992320} a^{35} - \frac{1}{1188717093793129655992320} a^{34} + \frac{1}{198119515632188275998720} a^{33} - \frac{17}{1188717093793129655992320} a^{32} + \frac{17}{1188717093793129655992320} a^{31} + \frac{2660043871}{198119515632188275998720} a^{30} - \frac{3929121828503}{1188717093793129655992320} a^{29} + \frac{4008923145143}{1188717093793129655992320} a^{28} - \frac{3958382311271}{198119515632188275998720} a^{27} + \frac{66795071088223}{1188717093793129655992320} a^{26} - \frac{68151693471103}{1188717093793129655992320} a^{25} - \frac{69642322758932639}{1188717093793129655992320} a^{24} + \frac{11625967407380387}{198119515632188275998720} a^{23} - \frac{419963130637294927}{1188717093793129655992320} a^{22} + \frac{26357076383106951633919}{1188717093793129655992320} a^{21} + \frac{47574852646921577416913}{198119515632188275998720} a^{20} - \frac{113287765418624332159033}{1188717093793129655992320} a^{19} + \frac{567453721225164569627161}{1188717093793129655992320} a^{18} - \frac{38215118669987384712553}{198119515632188275998720} a^{17} - \frac{586296792111919292726287}{1188717093793129655992320} a^{16} + \frac{113480459731786291311919}{1188717093793129655992320} a^{15} + \frac{81345111631950341997473}{198119515632188275998720} a^{14} - \frac{2668946942783807103973}{5503319878671896555520} a^{13} - \frac{1535621433446313588839}{5503319878671896555520} a^{12} - \frac{42247135286720284183}{917219979778649425920} a^{11} + \frac{43186634340488953673}{152869996629774904320} a^{10} + \frac{7782492811628491759}{25478332771629150720} a^{9} - \frac{46824182846006969}{235910488626195840} a^{8} + \frac{140139944832107327}{707731465878587520} a^{7} - \frac{22808026075580279}{117955244313097920} a^{6} + \frac{412767828890339}{1092178188084240} a^{5} - \frac{1269442529653597}{3276534564252720} a^{4} + \frac{168629737877509}{546089094042120} a^{3} + \frac{726373696042}{7584570750585} a^{2} + \frac{422973765376}{2528190250195} a + \frac{1027499163912}{2528190250195}$, $\frac{1}{7132302562758777935953920} a^{36} - \frac{1}{7132302562758777935953920} a^{35} + \frac{1}{1188717093793129655992320} a^{34} - \frac{17}{7132302562758777935953920} a^{33} + \frac{17}{7132302562758777935953920} a^{32} - \frac{17}{1188717093793129655992320} a^{31} - \frac{1939306515191}{7132302562758777935953920} a^{30} - \frac{21472376059657}{7132302562758777935953920} a^{29} + \frac{1962640580617}{1188717093793129655992320} a^{28} - \frac{107501884687169}{7132302562758777935953920} a^{27} + \frac{365030393010497}{7132302562758777935953920} a^{26} - \frac{70242819677026367}{7132302562758777935953920} a^{25} + \frac{12048175430798099}{1188717093793129655992320} a^{24} - \frac{421823265479245327}{7132302562758777935953920} a^{23} + \frac{1191584352316968607}{7132302562758777935953920} a^{22} + \frac{51184263753846955636961}{1188717093793129655992320} a^{21} + \frac{699515055822035227824167}{7132302562758777935953920} a^{20} - \frac{2979523014871927971127847}{7132302562758777935953920} a^{19} - \frac{567830460594029674981081}{1188717093793129655992320} a^{18} + \frac{1785686634476496153534353}{7132302562758777935953920} a^{17} + \frac{2403205878376024965945967}{7132302562758777935953920} a^{16} - \frac{213966046939308286912879}{1188717093793129655992320} a^{15} + \frac{10455208695875124191327}{33019919272031379333120} a^{14} - \frac{8303841874041442241567}{33019919272031379333120} a^{13} - \frac{1185691072958883773761}{5503319878671896555520} a^{12} - \frac{1961155511383987157}{4246388795271525120} a^{11} + \frac{1499171262474632461}{12739166385814575360} a^{10} + \frac{1362718299448874897}{3184791596453643840} a^{9} - \frac{276830029416496057}{1061597198817881280} a^{8} + \frac{23588652370577119}{58977622156548960} a^{7} + \frac{14518847591289331}{29488811078274480} a^{6} - \frac{3260859546365021}{9829603692758160} a^{5} - \frac{119392997822933}{546089094042120} a^{4} - \frac{47963064145007}{136522273510530} a^{3} - \frac{10831907219386}{22753712251755} a^{2} - \frac{396876624241}{15169141501170} a - \frac{949039810578}{2528190250195}$, $\frac{1}{42793815376552667615723520} a^{37} - \frac{1}{42793815376552667615723520} a^{36} + \frac{1}{7132302562758777935953920} a^{35} - \frac{17}{42793815376552667615723520} a^{34} + \frac{17}{42793815376552667615723520} a^{33} - \frac{17}{7132302562758777935953920} a^{32} + \frac{73}{42793815376552667615723520} a^{31} - \frac{11578354513993}{42793815376552667615723520} a^{30} - \frac{19446888727223}{7132302562758777935953920} a^{29} + \frac{58789559796607}{42793815376552667615723520} a^{28} - \frac{572726094531967}{42793815376552667615723520} a^{27} - \frac{68059047687626687}{42793815376552667615723520} a^{26} + \frac{11507201303544563}{7132302562758777935953920} a^{25} - \frac{408018513844886863}{42793815376552667615723520} a^{24} + \frac{1182206978480134687}{42793815376552667615723520} a^{23} - \frac{197738846312935423}{7132302562758777935953920} a^{22} - \frac{2310695430658704762202777}{42793815376552667615723520} a^{21} - \frac{4075808264709703671067367}{42793815376552667615723520} a^{20} + \frac{2584216521095281597302503}{7132302562758777935953920} a^{19} + \frac{3207811909379406200721617}{42793815376552667615723520} a^{18} + \frac{10783307309239964721163567}{42793815376552667615723520} a^{17} + \frac{2693206605203932220376017}{7132302562758777935953920} a^{16} + \frac{69026111516409682300583}{198119515632188275998720} a^{15} + \frac{65217275949690122817433}{198119515632188275998720} a^{14} + \frac{2503797591154148624903}{33019919272031379333120} a^{13} - \frac{104752135417725591943}{917219979778649425920} a^{12} - \frac{22361400247223375633}{917219979778649425920} a^{11} + \frac{24021126978771922439}{50956665543258301440} a^{10} + \frac{256177908358399333}{707731465878587520} a^{9} - \frac{7011816609502139}{1415462931757175040} a^{8} - \frac{75233086212696739}{235910488626195840} a^{7} + \frac{2984350813087237}{19659207385516320} a^{6} + \frac{1654350135586589}{6553069128505440} a^{5} - \frac{540377386372429}{1092178188084240} a^{4} + \frac{29744028342007}{91014849007020} a^{3} + \frac{6664301308753}{30338283002340} a^{2} + \frac{2234592350983}{7584570750585} a - \frac{873195350961}{2528190250195}$, $\frac{1}{256762892259316005694341120} a^{38} - \frac{1}{256762892259316005694341120} a^{37} + \frac{1}{42793815376552667615723520} a^{36} - \frac{17}{256762892259316005694341120} a^{35} + \frac{17}{256762892259316005694341120} a^{34} - \frac{17}{42793815376552667615723520} a^{33} + \frac{73}{256762892259316005694341120} a^{32} - \frac{73}{256762892259316005694341120} a^{31} - \frac{2061090105911}{42793815376552667615723520} a^{30} + \frac{417585930048127}{256762892259316005694341120} a^{29} - \frac{479418633227647}{256762892259316005694341120} a^{28} - \frac{67401082810541759}{256762892259316005694341120} a^{27} + \frac{10490611587831923}{42793815376552667615723520} a^{26} - \frac{412105665262067023}{256762892259316005694341120} a^{25} + \frac{1148489580556470559}{256762892259316005694341120} a^{24} - \frac{193373490474875263}{42793815376552667615723520} a^{23} + \frac{7109350734232311143}{256762892259316005694341120} a^{22} - \frac{10125365117049635305695719}{256762892259316005694341120} a^{21} - \frac{16406127644925176213442457}{42793815376552667615723520} a^{20} - \frac{21661365317900369731037743}{256762892259316005694341120} a^{19} - \frac{114506160122494189148662481}{256762892259316005694341120} a^{18} - \frac{10953082678863077932486063}{42793815376552667615723520} a^{17} - \frac{7372666216244244960697}{1188717093793129655992320} a^{16} + \frac{522197030920976117448601}{1188717093793129655992320} a^{15} + \frac{23770473956801614344983}{198119515632188275998720} a^{14} + \frac{1804597015278648553277}{5503319878671896555520} a^{13} - \frac{773197920143260330001}{5503319878671896555520} a^{12} + \frac{225232838386443799487}{917219979778649425920} a^{11} + \frac{14311723475148132883}{152869996629774904320} a^{10} + \frac{11160635054586351011}{25478332771629150720} a^{9} - \frac{58056474781208497}{117955244313097920} a^{8} + \frac{348498302816841607}{707731465878587520} a^{7} + \frac{5666259727518419}{117955244313097920} a^{6} + \frac{226852560493853}{546089094042120} a^{5} - \frac{1346186939413463}{3276534564252720} a^{4} + \frac{258148281924317}{546089094042120} a^{3} + \frac{1270008728177}{5056380500390} a^{2} - \frac{2861580985633}{7584570750585} a - \frac{1042485312378}{2528190250195}$, $\frac{1}{1540577353555896034166046720} a^{39} - \frac{1}{1540577353555896034166046720} a^{38} + \frac{1}{256762892259316005694341120} a^{37} - \frac{17}{1540577353555896034166046720} a^{36} + \frac{17}{1540577353555896034166046720} a^{35} - \frac{17}{256762892259316005694341120} a^{34} + \frac{73}{1540577353555896034166046720} a^{33} - \frac{73}{1540577353555896034166046720} a^{32} + \frac{73}{256762892259316005694341120} a^{31} + \frac{413201720782207}{1540577353555896034166046720} a^{30} + \frac{10740723179800193}{1540577353555896034166046720} a^{29} - \frac{78717344913714623}{1540577353555896034166046720} a^{28} + \frac{21656958414667763}{256762892259316005694341120} a^{27} - \frac{602848076083540303}{1540577353555896034166046720} a^{26} + \frac{1338194863533053983}{1540577353555896034166046720} a^{25} - \frac{383043554997511423}{256762892259316005694341120} a^{24} + \frac{7928421086583343463}{1540577353555896034166046720} a^{23} - \frac{5746366178699559143}{1540577353555896034166046720} a^{22} + \frac{12586915706205547338498023}{256762892259316005694341120} a^{21} - \frac{670253168271576957658547503}{1540577353555896034166046720} a^{20} - \frac{370348595082102032380006097}{1540577353555896034166046720} a^{19} - \frac{37773132002641438871764783}{256762892259316005694341120} a^{18} - \frac{707932044023826356760217}{7132302562758777935953920} a^{17} + \frac{3341145059282173380732697}{7132302562758777935953920} a^{16} - \frac{508540082795510229915097}{1188717093793129655992320} a^{15} - \frac{11171449096577204296783}{33019919272031379333120} a^{14} - \frac{1260109662412704974177}{33019919272031379333120} a^{13} + \frac{270123852930815725097}{5503319878671896555520} a^{12} + \frac{29207677411072165793}{152869996629774904320} a^{11} - \frac{60211780928021333423}{152869996629774904320} a^{10} - \frac{421415135714216231}{2123194397635762560} a^{9} + \frac{838424953856183177}{4246388795271525120} a^{8} - \frac{134225171595894347}{707731465878587520} a^{7} + \frac{1804670138485999}{9829603692758160} a^{6} - \frac{3365632295249713}{19659207385516320} a^{5} + \frac{390290038496083}{3276534564252720} a^{4} - \frac{11335205325797}{45507424503510} a^{3} + \frac{23884678628309}{91014849007020} a^{2} - \frac{900958631459}{15169141501170} a - \frac{213464336940}{505638050039}$

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:  \( -\frac{218264857259}{308115470711179206833209344} a^{39} + \frac{2400913429849}{1540577353555896034166046720} a^{38} + \frac{3710502573403}{308115470711179206833209344} a^{36} + \frac{6329680860511}{1540577353555896034166046720} a^{35} - \frac{15933334579907}{308115470711179206833209344} a^{33} - \frac{626201875476071}{1540577353555896034166046720} a^{32} + \frac{43229811565271}{1540577353555896034166046720} a^{30} + \frac{9278220817222831}{1540577353555896034166046720} a^{29} + \frac{6410220592839571}{308115470711179206833209344} a^{28} - \frac{6329680860511}{1188717093793129655992320} a^{27} - \frac{22470148789956791}{1540577353555896034166046720} a^{26} - \frac{108973750078272707}{308115470711179206833209344} a^{25} - \frac{2400913429849}{5503319878671896555520} a^{24} - \frac{1622103167090866049}{1540577353555896034166046720} a^{23} + \frac{467946103277288683}{308115470711179206833209344} a^{22} + \frac{218264857259}{25478332771629150720} a^{21} + \frac{6410220592839571}{304522109815357982638080} a^{20} - \frac{3961389797343948563}{308115470711179206833209344} a^{19} - \frac{218264857259}{4246388795271525120} a^{18} - \frac{3455108899540528769}{7132302562758777935953920} a^{17} - \frac{97241140941172421}{237743418758625931198464} a^{16} - \frac{4147032287921}{4246388795271525120} a^{15} + \frac{121794191263951849}{33019919272031379333120} a^{14} + \frac{12461832025202605}{1100663975734379311104} a^{13} + \frac{117644758062601}{4246388795271525120} a^{12} + \frac{6410220592839571}{152869996629774904320} a^{11} - \frac{530601867996629}{5095666554325830144} a^{10} - \frac{218264857259}{839373155815680} a^{9} - \frac{1605271176}{2528190250195} a^{8} - \frac{15933334579907}{23591048862619584} a^{7} + \frac{117644758062601}{19659207385516320} a^{6} + \frac{3710502573403}{109217818808424} a^{4} - \frac{4147032287921}{91014849007020} a^{3} - \frac{218264857259}{505638050039} a - \frac{1309589143554}{2528190250195} \) (order $66$)
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

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-759}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{253}) \), \(\Q(\sqrt{-23}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{-23})\), \(\Q(\sqrt{-11}, \sqrt{-23})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{33}, \sqrt{69})\), \(\Q(\sqrt{-3}, \sqrt{253})\), \(\Q(\sqrt{-11}, \sqrt{69})\), \(\Q(\zeta_{11})^+\), 8.0.331869318561.5, 10.0.1379687283212183.1, \(\Q(\zeta_{33})^+\), 10.0.3687904108026165159.1, 10.0.52089208083.1, 10.10.335264009820560469.1, \(\Q(\zeta_{11})\), 10.10.15176560115334013.1, 20.0.13600636709996264858725830959545495281.1, 20.0.112401956280960866601039925285499961.1, 20.0.230327976934347149972494554684169.1, \(\Q(\zeta_{33})\), 20.20.13600636709996264858725830959545495281.1, 20.0.13600636709996264858725830959545495281.2, 20.0.13600636709996264858725830959545495281.3

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$