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Properties

Label 40.40.2356601762...5625.1
Degree $40$
Signature $[40, 0]$
Discriminant $3^{20}\cdot 5^{70}\cdot 7^{20}$
Root discriminant $76.61$
Ramified primes $3, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3248255, 46652100, 60864000, -1531206050, 1748896000, 11028952605, -17934217600, -37349092000, 71681548800, 73797867000, -161908455840, -94254254600, 236443720400, 82634574550, -239841990400, -51717568485, 176698891480, 23720246000, -97386692440, -8107946000, 40956171311, 2084212900, -13309824000, -403616950, 3365024000, 58522739, -662762880, -6252400, 101261440, 477400, -11872224, -24640, 1047200, 770, -67200, -11, 2960, 0, -80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 80*x^38 + 2960*x^36 - 11*x^35 - 67200*x^34 + 770*x^33 + 1047200*x^32 - 24640*x^31 - 11872224*x^30 + 477400*x^29 + 101261440*x^28 - 6252400*x^27 - 662762880*x^26 + 58522739*x^25 + 3365024000*x^24 - 403616950*x^23 - 13309824000*x^22 + 2084212900*x^21 + 40956171311*x^20 - 8107946000*x^19 - 97386692440*x^18 + 23720246000*x^17 + 176698891480*x^16 - 51717568485*x^15 - 239841990400*x^14 + 82634574550*x^13 + 236443720400*x^12 - 94254254600*x^11 - 161908455840*x^10 + 73797867000*x^9 + 71681548800*x^8 - 37349092000*x^7 - 17934217600*x^6 + 11028952605*x^5 + 1748896000*x^4 - 1531206050*x^3 + 60864000*x^2 + 46652100*x - 3248255)
 
gp: K = bnfinit(x^40 - 80*x^38 + 2960*x^36 - 11*x^35 - 67200*x^34 + 770*x^33 + 1047200*x^32 - 24640*x^31 - 11872224*x^30 + 477400*x^29 + 101261440*x^28 - 6252400*x^27 - 662762880*x^26 + 58522739*x^25 + 3365024000*x^24 - 403616950*x^23 - 13309824000*x^22 + 2084212900*x^21 + 40956171311*x^20 - 8107946000*x^19 - 97386692440*x^18 + 23720246000*x^17 + 176698891480*x^16 - 51717568485*x^15 - 239841990400*x^14 + 82634574550*x^13 + 236443720400*x^12 - 94254254600*x^11 - 161908455840*x^10 + 73797867000*x^9 + 71681548800*x^8 - 37349092000*x^7 - 17934217600*x^6 + 11028952605*x^5 + 1748896000*x^4 - 1531206050*x^3 + 60864000*x^2 + 46652100*x - 3248255, 1)
 

Normalized defining polynomial

\( x^{40} - 80 x^{38} + 2960 x^{36} - 11 x^{35} - 67200 x^{34} + 770 x^{33} + 1047200 x^{32} - 24640 x^{31} - 11872224 x^{30} + 477400 x^{29} + 101261440 x^{28} - 6252400 x^{27} - 662762880 x^{26} + 58522739 x^{25} + 3365024000 x^{24} - 403616950 x^{23} - 13309824000 x^{22} + 2084212900 x^{21} + 40956171311 x^{20} - 8107946000 x^{19} - 97386692440 x^{18} + 23720246000 x^{17} + 176698891480 x^{16} - 51717568485 x^{15} - 239841990400 x^{14} + 82634574550 x^{13} + 236443720400 x^{12} - 94254254600 x^{11} - 161908455840 x^{10} + 73797867000 x^{9} + 71681548800 x^{8} - 37349092000 x^{7} - 17934217600 x^{6} + 11028952605 x^{5} + 1748896000 x^{4} - 1531206050 x^{3} + 60864000 x^{2} + 46652100 x - 3248255 \)

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:  \(2356601762749139913517942415268160462338276062155273393727838993072509765625=3^{20}\cdot 5^{70}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.61$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(525=3\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{525}(512,·)$, $\chi_{525}(1,·)$, $\chi_{525}(517,·)$, $\chi_{525}(8,·)$, $\chi_{525}(524,·)$, $\chi_{525}(13,·)$, $\chi_{525}(274,·)$, $\chi_{525}(407,·)$, $\chi_{525}(412,·)$, $\chi_{525}(419,·)$, $\chi_{525}(421,·)$, $\chi_{525}(41,·)$, $\chi_{525}(428,·)$, $\chi_{525}(302,·)$, $\chi_{525}(433,·)$, $\chi_{525}(307,·)$, $\chi_{525}(314,·)$, $\chi_{525}(316,·)$, $\chi_{525}(64,·)$, $\chi_{525}(323,·)$, $\chi_{525}(197,·)$, $\chi_{525}(328,·)$, $\chi_{525}(202,·)$, $\chi_{525}(461,·)$, $\chi_{525}(209,·)$, $\chi_{525}(211,·)$, $\chi_{525}(356,·)$, $\chi_{525}(218,·)$, $\chi_{525}(92,·)$, $\chi_{525}(223,·)$, $\chi_{525}(97,·)$, $\chi_{525}(251,·)$, $\chi_{525}(484,·)$, $\chi_{525}(104,·)$, $\chi_{525}(106,·)$, $\chi_{525}(146,·)$, $\chi_{525}(113,·)$, $\chi_{525}(118,·)$, $\chi_{525}(169,·)$, $\chi_{525}(379,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{4049} a^{25} - \frac{50}{4049} a^{23} + \frac{1100}{4049} a^{21} - \frac{1853}{4049} a^{19} + \frac{628}{4049} a^{17} - \frac{663}{4049} a^{15} + \frac{1164}{4049} a^{13} - \frac{1147}{4049} a^{11} + \frac{1260}{4049} a^{9} - \frac{1260}{4049} a^{7} - \frac{914}{4049} a^{5} + \frac{921}{4049} a^{3} + \frac{1175}{4049} a + \frac{181}{4049}$, $\frac{1}{4049} a^{26} - \frac{50}{4049} a^{24} + \frac{1100}{4049} a^{22} - \frac{1853}{4049} a^{20} + \frac{628}{4049} a^{18} - \frac{663}{4049} a^{16} + \frac{1164}{4049} a^{14} - \frac{1147}{4049} a^{12} + \frac{1260}{4049} a^{10} - \frac{1260}{4049} a^{8} - \frac{914}{4049} a^{6} + \frac{921}{4049} a^{4} + \frac{1175}{4049} a^{2} + \frac{181}{4049} a$, $\frac{1}{4049} a^{27} - \frac{1400}{4049} a^{23} + \frac{510}{4049} a^{21} + \frac{1105}{4049} a^{19} - \frac{1655}{4049} a^{17} + \frac{406}{4049} a^{15} + \frac{367}{4049} a^{13} + \frac{596}{4049} a^{11} + \frac{1005}{4049} a^{9} + \frac{870}{4049} a^{7} - \frac{240}{4049} a^{5} - \frac{1363}{4049} a^{3} + \frac{181}{4049} a^{2} - \frac{1985}{4049} a + \frac{952}{4049}$, $\frac{1}{4049} a^{28} - \frac{1400}{4049} a^{24} + \frac{510}{4049} a^{22} + \frac{1105}{4049} a^{20} - \frac{1655}{4049} a^{18} + \frac{406}{4049} a^{16} + \frac{367}{4049} a^{14} + \frac{596}{4049} a^{12} + \frac{1005}{4049} a^{10} + \frac{870}{4049} a^{8} - \frac{240}{4049} a^{6} - \frac{1363}{4049} a^{4} + \frac{181}{4049} a^{3} - \frac{1985}{4049} a^{2} + \frac{952}{4049} a$, $\frac{1}{4049} a^{29} - \frac{657}{4049} a^{23} - \frac{1564}{4049} a^{21} - \frac{446}{4049} a^{19} + \frac{973}{4049} a^{17} - \frac{612}{4049} a^{15} - \frac{1551}{4049} a^{13} - \frac{1391}{4049} a^{11} - \frac{494}{4049} a^{9} + \frac{1124}{4049} a^{7} - \frac{1479}{4049} a^{5} + \frac{181}{4049} a^{4} - \frac{167}{4049} a^{3} + \frac{952}{4049} a^{2} + \frac{1106}{4049} a - \frac{1687}{4049}$, $\frac{1}{360361} a^{30} + \frac{29}{360361} a^{28} + \frac{18}{360361} a^{26} - \frac{70500}{360361} a^{24} + \frac{634}{360361} a^{22} - \frac{155617}{360361} a^{20} - \frac{31669}{360361} a^{18} - \frac{174879}{360361} a^{16} - \frac{8}{89} a^{15} - \frac{46887}{360361} a^{14} - \frac{27}{89} a^{13} + \frac{60031}{360361} a^{12} - \frac{32}{89} a^{11} - \frac{78237}{360361} a^{10} - \frac{22}{89} a^{9} + \frac{133242}{360361} a^{8} - \frac{17}{89} a^{7} + \frac{7501}{360361} a^{6} + \frac{125700}{360361} a^{5} + \frac{1178}{360361} a^{4} - \frac{9995}{360361} a^{3} - \frac{2917}{360361} a^{2} + \frac{134453}{360361} a + \frac{32}{89}$, $\frac{1}{360361} a^{31} + \frac{29}{360361} a^{29} + \frac{18}{360361} a^{27} - \frac{12}{360361} a^{25} + \frac{79844}{360361} a^{23} - \frac{96432}{360361} a^{21} + \frac{165110}{360361} a^{19} + \frac{127543}{360361} a^{17} - \frac{8}{89} a^{16} + \frac{66499}{360361} a^{15} - \frac{27}{89} a^{14} - \frac{54245}{360361} a^{13} - \frac{32}{89} a^{12} + \frac{153252}{360361} a^{11} - \frac{22}{89} a^{10} - \frac{61045}{360361} a^{9} - \frac{17}{89} a^{8} - \frac{158573}{360361} a^{7} + \frac{125700}{360361} a^{6} + \frac{79765}{360361} a^{5} - \frac{9995}{360361} a^{4} + \frac{51551}{360361} a^{3} + \frac{134453}{360361} a^{2} + \frac{69938}{360361} a + \frac{1637}{4049}$, $\frac{1}{360361} a^{32} - \frac{22}{360361} a^{28} - \frac{104839}{360361} a^{24} + \frac{160370}{360361} a^{22} - \frac{111087}{360361} a^{20} + \frac{55641}{360361} a^{18} - \frac{8}{89} a^{17} + \frac{64100}{360361} a^{16} + \frac{27}{89} a^{15} + \frac{58855}{360361} a^{14} + \frac{39}{89} a^{13} + \frac{79056}{360361} a^{12} + \frac{16}{89} a^{11} + \frac{82063}{360361} a^{10} - \frac{2}{89} a^{9} - \frac{34590}{360361} a^{8} - \frac{40309}{360361} a^{7} - \frac{97358}{360361} a^{6} - \frac{51685}{360361} a^{5} + \frac{138162}{360361} a^{4} - \frac{151433}{360361} a^{3} - \frac{87282}{360361} a^{2} - \frac{11350}{360361} a - \frac{38}{89}$, $\frac{1}{57080822039} a^{33} - \frac{44742}{57080822039} a^{32} - \frac{66}{57080822039} a^{31} + \frac{12306}{57080822039} a^{30} + \frac{1980}{57080822039} a^{29} + \frac{1341732}{57080822039} a^{28} + \frac{4259598}{57080822039} a^{27} + \frac{1209319}{57080822039} a^{26} - \frac{5954996}{57080822039} a^{25} + \frac{19896267250}{57080822039} a^{24} + \frac{26732435049}{57080822039} a^{23} - \frac{23260458410}{57080822039} a^{22} - \frac{9826167357}{57080822039} a^{21} - \frac{28185094615}{57080822039} a^{20} + \frac{9269990360}{57080822039} a^{19} + \frac{27360813927}{57080822039} a^{18} - \frac{24917681559}{57080822039} a^{17} + \frac{11890075828}{57080822039} a^{16} + \frac{15476016533}{57080822039} a^{15} - \frac{20868429514}{57080822039} a^{14} - \frac{27864863723}{57080822039} a^{13} - \frac{15483207988}{57080822039} a^{12} + \frac{25457309783}{57080822039} a^{11} + \frac{1967602801}{57080822039} a^{10} - \frac{4413992209}{57080822039} a^{9} - \frac{5476890462}{57080822039} a^{8} - \frac{17943991490}{57080822039} a^{7} - \frac{4430257706}{57080822039} a^{6} - \frac{14931290752}{57080822039} a^{5} - \frac{16092219162}{57080822039} a^{4} + \frac{2904459983}{57080822039} a^{3} - \frac{8867070380}{57080822039} a^{2} - \frac{27211827264}{57080822039} a - \frac{25175077736}{57080822039}$, $\frac{1}{57080822039} a^{34} - \frac{68}{57080822039} a^{32} + \frac{68915}{57080822039} a^{31} + \frac{2108}{57080822039} a^{30} - \frac{6058902}{57080822039} a^{29} - \frac{39440}{57080822039} a^{28} - \frac{2325760}{57080822039} a^{27} - \frac{5009915}{57080822039} a^{26} + \frac{1286147}{57080822039} a^{25} - \frac{24379707840}{57080822039} a^{24} - \frac{26524391467}{57080822039} a^{23} - \frac{21496948999}{57080822039} a^{22} - \frac{9214181273}{57080822039} a^{21} + \frac{2229460206}{57080822039} a^{20} - \frac{22851789081}{57080822039} a^{19} + \frac{20474020040}{57080822039} a^{18} - \frac{16020278370}{57080822039} a^{17} + \frac{25320436802}{57080822039} a^{16} + \frac{10872377642}{57080822039} a^{15} + \frac{13174394921}{57080822039} a^{14} + \frac{25565260196}{57080822039} a^{13} - \frac{23973253431}{57080822039} a^{12} + \frac{19560178640}{57080822039} a^{11} + \frac{4530730433}{57080822039} a^{10} + \frac{43073948}{378018689} a^{9} - \frac{5944175515}{57080822039} a^{8} + \frac{26460624597}{57080822039} a^{7} - \frac{9639865899}{57080822039} a^{6} + \frac{7066475565}{57080822039} a^{5} - \frac{6279144099}{57080822039} a^{4} - \frac{610190014}{57080822039} a^{3} + \frac{3370186386}{57080822039} a^{2} - \frac{6438858485}{57080822039} a - \frac{26584419032}{57080822039}$, $\frac{1}{57080822039} a^{35} + \frac{36040}{57080822039} a^{32} - \frac{2380}{57080822039} a^{31} + \frac{57}{641357551} a^{30} + \frac{95200}{57080822039} a^{29} + \frac{5118945}{57080822039} a^{28} + \frac{2692529}{57080822039} a^{27} - \frac{63582}{641357551} a^{26} - \frac{1223230}{57080822039} a^{25} - \frac{24906540100}{57080822039} a^{24} - \frac{18506249230}{57080822039} a^{23} - \frac{22896269989}{57080822039} a^{22} + \frac{23601635473}{57080822039} a^{21} - \frac{23415723879}{57080822039} a^{20} + \frac{23169882267}{57080822039} a^{19} + \frac{13487730655}{57080822039} a^{18} - \frac{6155668348}{57080822039} a^{17} - \frac{20229154147}{57080822039} a^{16} - \frac{22980269598}{57080822039} a^{15} - \frac{24429549856}{57080822039} a^{14} + \frac{15617865195}{57080822039} a^{13} + \frac{27367875291}{57080822039} a^{12} + \frac{410627200}{57080822039} a^{11} - \frac{23048920927}{57080822039} a^{10} + \frac{8987843497}{57080822039} a^{9} - \frac{12336865089}{57080822039} a^{8} - \frac{19399826480}{57080822039} a^{7} + \frac{2603533842}{57080822039} a^{6} + \frac{13248916343}{57080822039} a^{5} + \frac{16764672960}{57080822039} a^{4} - \frac{17319414881}{57080822039} a^{3} - \frac{18767789085}{57080822039} a^{2} + \frac{22627332815}{57080822039} a - \frac{20188126148}{57080822039}$, $\frac{1}{57080822039} a^{36} - \frac{2520}{57080822039} a^{32} + \frac{7728}{57080822039} a^{31} - \frac{54239}{57080822039} a^{30} + \frac{5831290}{57080822039} a^{29} - \frac{6785971}{57080822039} a^{28} - \frac{151125}{57080822039} a^{27} + \frac{5691214}{57080822039} a^{26} + \frac{1627011}{57080822039} a^{25} - \frac{1178900430}{57080822039} a^{24} - \frac{1567807433}{57080822039} a^{23} - \frac{19528345555}{57080822039} a^{22} + \frac{26045095870}{57080822039} a^{21} - \frac{13394157043}{57080822039} a^{20} - \frac{26905128551}{57080822039} a^{19} + \frac{13645589364}{57080822039} a^{18} - \frac{301691636}{641357551} a^{17} + \frac{5461176142}{57080822039} a^{16} + \frac{8134497242}{57080822039} a^{15} + \frac{17420685444}{57080822039} a^{14} - \frac{28049216951}{57080822039} a^{13} - \frac{9140271536}{57080822039} a^{12} - \frac{523477458}{57080822039} a^{11} - \frac{3389524553}{57080822039} a^{10} + \frac{21600165253}{57080822039} a^{9} - \frac{12812256436}{57080822039} a^{8} - \frac{5817602024}{57080822039} a^{7} - \frac{28163083309}{57080822039} a^{6} - \frac{21402315259}{57080822039} a^{5} + \frac{19303460936}{57080822039} a^{4} + \frac{16369692403}{57080822039} a^{3} + \frac{10701484869}{57080822039} a^{2} - \frac{13793060604}{57080822039} a + \frac{20307362517}{57080822039}$, $\frac{1}{57080822039} a^{37} + \frac{37976}{57080822039} a^{32} - \frac{62160}{57080822039} a^{31} - \frac{64557}{57080822039} a^{30} + \frac{2797200}{57080822039} a^{29} - \frac{1280332}{57080822039} a^{28} + \frac{425974}{57080822039} a^{27} + \frac{2306126}{57080822039} a^{26} - \frac{3448510}{57080822039} a^{25} - \frac{15286784500}{57080822039} a^{24} + \frac{20388097350}{57080822039} a^{23} - \frac{22677112022}{57080822039} a^{22} - \frac{23229258569}{57080822039} a^{21} - \frac{2031786244}{57080822039} a^{20} - \frac{21810852743}{57080822039} a^{19} - \frac{14403856118}{57080822039} a^{18} - \frac{5564157497}{57080822039} a^{17} - \frac{24453964170}{57080822039} a^{16} + \frac{16115366153}{57080822039} a^{15} + \frac{716797526}{57080822039} a^{14} + \frac{8934036949}{57080822039} a^{13} + \frac{5963669709}{57080822039} a^{12} - \frac{19440809646}{57080822039} a^{11} + \frac{6298576944}{57080822039} a^{10} + \frac{25184867555}{57080822039} a^{9} - \frac{23443663086}{57080822039} a^{8} - \frac{13317473426}{57080822039} a^{7} - \frac{21946820759}{57080822039} a^{6} + \frac{218184823}{57080822039} a^{5} - \frac{25207483547}{57080822039} a^{4} - \frac{456199921}{57080822039} a^{3} + \frac{6515060868}{57080822039} a^{2} - \frac{10159863785}{57080822039} a + \frac{20468184132}{57080822039}$, $\frac{1}{57080822039} a^{38} + \frac{72358}{57080822039} a^{32} + \frac{65874}{57080822039} a^{31} + \frac{48811}{57080822039} a^{30} - \frac{4401267}{57080822039} a^{29} - \frac{3833310}{57080822039} a^{28} - \frac{5821505}{57080822039} a^{27} + \frac{3836767}{57080822039} a^{26} + \frac{3988367}{57080822039} a^{25} - \frac{18111637939}{57080822039} a^{24} - \frac{6058550420}{57080822039} a^{23} - \frac{26943473589}{57080822039} a^{22} + \frac{8798562897}{57080822039} a^{21} + \frac{22727380439}{57080822039} a^{20} + \frac{27103890066}{57080822039} a^{19} + \frac{16190447552}{57080822039} a^{18} + \frac{17517321187}{57080822039} a^{17} + \frac{11338251333}{57080822039} a^{16} + \frac{10002634388}{57080822039} a^{15} - \frac{7231105270}{57080822039} a^{14} + \frac{1545463137}{57080822039} a^{13} - \frac{18176313090}{57080822039} a^{12} - \frac{21715344073}{57080822039} a^{11} - \frac{11187018780}{57080822039} a^{10} + \frac{12036688871}{57080822039} a^{9} - \frac{21132029032}{57080822039} a^{8} - \frac{10591505601}{57080822039} a^{7} + \frac{6093993387}{57080822039} a^{6} + \frac{19959519110}{57080822039} a^{5} + \frac{7446139566}{57080822039} a^{4} + \frac{24511321904}{57080822039} a^{3} + \frac{1037189959}{57080822039} a^{2} - \frac{139353484}{378018689} a - \frac{12895113637}{57080822039}$, $\frac{1}{57080822039} a^{39} - \frac{9651}{57080822039} a^{32} + \frac{72469}{57080822039} a^{31} - \frac{42864}{57080822039} a^{30} - \frac{2959060}{57080822039} a^{29} - \frac{990506}{57080822039} a^{28} + \frac{777282}{57080822039} a^{27} - \frac{3793013}{57080822039} a^{26} + \frac{5474145}{57080822039} a^{25} + \frac{12795477058}{57080822039} a^{24} - \frac{22872675968}{57080822039} a^{23} + \frac{6458708556}{57080822039} a^{22} - \frac{19072516089}{57080822039} a^{21} + \frac{6711246465}{57080822039} a^{20} + \frac{8989622302}{57080822039} a^{19} + \frac{11243770615}{57080822039} a^{18} - \frac{26730634182}{57080822039} a^{17} - \frac{17433798609}{57080822039} a^{16} - \frac{9574665850}{57080822039} a^{15} - \frac{5024451454}{57080822039} a^{14} - \frac{16605515660}{57080822039} a^{13} + \frac{16011913531}{57080822039} a^{12} - \frac{27003712795}{57080822039} a^{11} - \frac{6833697120}{57080822039} a^{10} - \frac{10530110649}{57080822039} a^{9} - \frac{16775248981}{57080822039} a^{8} + \frac{23058526457}{57080822039} a^{7} - \frac{20233747686}{57080822039} a^{6} - \frac{11313631557}{57080822039} a^{5} + \frac{22096946665}{57080822039} a^{4} + \frac{23440309578}{57080822039} a^{3} + \frac{13403190278}{57080822039} a^{2} + \frac{23274660476}{57080822039} a + \frac{7870298975}{57080822039}$

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

$C_2\times C_{20}$ (as 40T2):

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 $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 4.4.6125.1, \(\Q(\zeta_{15})^+\), 5.5.390625.1, 8.8.3038765625.1, 10.10.3115921783447265625.1, \(\Q(\zeta_{25})^+\), 10.10.623184356689453125.1, 20.20.9708968560561188496649265289306640625.1, 20.20.822111175511963665485382080078125.1, \(\Q(\zeta_{75})^+\)

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 R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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
3Data not computed
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$