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Properties

Label 36.0.16699011555...8801.1
Degree $36$
Signature $[0, 18]$
Discriminant $11^{18}\cdot 19^{34}$
Root discriminant $53.51$
Ramified primes $11, 19$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![387420489, -129140163, -86093442, 71744535, 4782969, -25509168, 6908733, 6200145, -4369626, -610173, 1659933, -349920, -436671, 262197, 58158, -106785, 16209, 30192, -15467, 10064, 1801, -3955, 718, 1079, -599, -160, 253, -31, -74, 35, 13, -16, 1, 5, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 2*x^34 + 5*x^33 + x^32 - 16*x^31 + 13*x^30 + 35*x^29 - 74*x^28 - 31*x^27 + 253*x^26 - 160*x^25 - 599*x^24 + 1079*x^23 + 718*x^22 - 3955*x^21 + 1801*x^20 + 10064*x^19 - 15467*x^18 + 30192*x^17 + 16209*x^16 - 106785*x^15 + 58158*x^14 + 262197*x^13 - 436671*x^12 - 349920*x^11 + 1659933*x^10 - 610173*x^9 - 4369626*x^8 + 6200145*x^7 + 6908733*x^6 - 25509168*x^5 + 4782969*x^4 + 71744535*x^3 - 86093442*x^2 - 129140163*x + 387420489)
 
gp: K = bnfinit(x^36 - x^35 - 2*x^34 + 5*x^33 + x^32 - 16*x^31 + 13*x^30 + 35*x^29 - 74*x^28 - 31*x^27 + 253*x^26 - 160*x^25 - 599*x^24 + 1079*x^23 + 718*x^22 - 3955*x^21 + 1801*x^20 + 10064*x^19 - 15467*x^18 + 30192*x^17 + 16209*x^16 - 106785*x^15 + 58158*x^14 + 262197*x^13 - 436671*x^12 - 349920*x^11 + 1659933*x^10 - 610173*x^9 - 4369626*x^8 + 6200145*x^7 + 6908733*x^6 - 25509168*x^5 + 4782969*x^4 + 71744535*x^3 - 86093442*x^2 - 129140163*x + 387420489, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 2 x^{34} + 5 x^{33} + x^{32} - 16 x^{31} + 13 x^{30} + 35 x^{29} - 74 x^{28} - 31 x^{27} + 253 x^{26} - 160 x^{25} - 599 x^{24} + 1079 x^{23} + 718 x^{22} - 3955 x^{21} + 1801 x^{20} + 10064 x^{19} - 15467 x^{18} + 30192 x^{17} + 16209 x^{16} - 106785 x^{15} + 58158 x^{14} + 262197 x^{13} - 436671 x^{12} - 349920 x^{11} + 1659933 x^{10} - 610173 x^{9} - 4369626 x^{8} + 6200145 x^{7} + 6908733 x^{6} - 25509168 x^{5} + 4782969 x^{4} + 71744535 x^{3} - 86093442 x^{2} - 129140163 x + 387420489 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(166990115557548038315544519372094023948173088869511853538488801=11^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.51$
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:  $11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(209=11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{209}(1,·)$, $\chi_{209}(131,·)$, $\chi_{209}(10,·)$, $\chi_{209}(12,·)$, $\chi_{209}(142,·)$, $\chi_{209}(144,·)$, $\chi_{209}(21,·)$, $\chi_{209}(23,·)$, $\chi_{209}(153,·)$, $\chi_{209}(155,·)$, $\chi_{209}(32,·)$, $\chi_{209}(34,·)$, $\chi_{209}(164,·)$, $\chi_{209}(166,·)$, $\chi_{209}(43,·)$, $\chi_{209}(45,·)$, $\chi_{209}(175,·)$, $\chi_{209}(177,·)$, $\chi_{209}(54,·)$, $\chi_{209}(56,·)$, $\chi_{209}(186,·)$, $\chi_{209}(188,·)$, $\chi_{209}(65,·)$, $\chi_{209}(67,·)$, $\chi_{209}(197,·)$, $\chi_{209}(199,·)$, $\chi_{209}(78,·)$, $\chi_{209}(208,·)$, $\chi_{209}(87,·)$, $\chi_{209}(89,·)$, $\chi_{209}(98,·)$, $\chi_{209}(100,·)$, $\chi_{209}(109,·)$, $\chi_{209}(111,·)$, $\chi_{209}(120,·)$, $\chi_{209}(122,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{46401} a^{19} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{5403}{15467}$, $\frac{1}{139203} a^{20} - \frac{1}{139203} a^{19} - \frac{4}{9} a^{18} + \frac{1}{9} a^{17} + \frac{2}{9} a^{16} + \frac{4}{9} a^{15} - \frac{1}{9} a^{14} - \frac{2}{9} a^{13} - \frac{4}{9} a^{12} + \frac{1}{9} a^{11} + \frac{2}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{9} a^{8} - \frac{2}{9} a^{7} - \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2} + \frac{10064}{46401} a + \frac{1801}{15467}$, $\frac{1}{417609} a^{21} - \frac{1}{417609} a^{20} - \frac{2}{417609} a^{19} - \frac{8}{27} a^{18} - \frac{7}{27} a^{17} + \frac{4}{27} a^{16} - \frac{10}{27} a^{15} - \frac{2}{27} a^{14} + \frac{5}{27} a^{13} + \frac{1}{27} a^{12} + \frac{11}{27} a^{11} + \frac{13}{27} a^{10} + \frac{8}{27} a^{9} + \frac{7}{27} a^{8} - \frac{4}{27} a^{7} + \frac{10}{27} a^{6} + \frac{2}{27} a^{5} - \frac{5}{27} a^{4} - \frac{1}{27} a^{3} + \frac{10064}{139203} a^{2} + \frac{1801}{46401} a - \frac{3955}{15467}$, $\frac{1}{1252827} a^{22} - \frac{1}{1252827} a^{21} - \frac{2}{1252827} a^{20} + \frac{5}{1252827} a^{19} + \frac{20}{81} a^{18} + \frac{4}{81} a^{17} + \frac{17}{81} a^{16} - \frac{29}{81} a^{15} - \frac{22}{81} a^{14} + \frac{28}{81} a^{13} + \frac{38}{81} a^{12} + \frac{40}{81} a^{11} + \frac{8}{81} a^{10} + \frac{34}{81} a^{9} + \frac{23}{81} a^{8} + \frac{37}{81} a^{7} - \frac{25}{81} a^{6} - \frac{5}{81} a^{5} - \frac{1}{81} a^{4} + \frac{10064}{417609} a^{3} + \frac{1801}{139203} a^{2} - \frac{3955}{46401} a + \frac{718}{15467}$, $\frac{1}{3758481} a^{23} - \frac{1}{3758481} a^{22} - \frac{2}{3758481} a^{21} + \frac{5}{3758481} a^{20} + \frac{1}{3758481} a^{19} - \frac{77}{243} a^{18} + \frac{17}{243} a^{17} - \frac{29}{243} a^{16} - \frac{22}{243} a^{15} + \frac{109}{243} a^{14} - \frac{43}{243} a^{13} - \frac{41}{243} a^{12} - \frac{73}{243} a^{11} - \frac{47}{243} a^{10} + \frac{23}{243} a^{9} + \frac{118}{243} a^{8} + \frac{56}{243} a^{7} + \frac{76}{243} a^{6} - \frac{1}{243} a^{5} + \frac{10064}{1252827} a^{4} + \frac{1801}{417609} a^{3} - \frac{3955}{139203} a^{2} + \frac{718}{46401} a + \frac{1079}{15467}$, $\frac{1}{11275443} a^{24} - \frac{1}{11275443} a^{23} - \frac{2}{11275443} a^{22} + \frac{5}{11275443} a^{21} + \frac{1}{11275443} a^{20} - \frac{16}{11275443} a^{19} - \frac{226}{729} a^{18} - \frac{272}{729} a^{17} + \frac{221}{729} a^{16} - \frac{134}{729} a^{15} + \frac{200}{729} a^{14} + \frac{202}{729} a^{13} - \frac{73}{729} a^{12} + \frac{196}{729} a^{11} + \frac{23}{729} a^{10} + \frac{118}{729} a^{9} - \frac{187}{729} a^{8} - \frac{167}{729} a^{7} - \frac{1}{729} a^{6} + \frac{10064}{3758481} a^{5} + \frac{1801}{1252827} a^{4} - \frac{3955}{417609} a^{3} + \frac{718}{139203} a^{2} + \frac{1079}{46401} a - \frac{599}{15467}$, $\frac{1}{33826329} a^{25} - \frac{1}{33826329} a^{24} - \frac{2}{33826329} a^{23} + \frac{5}{33826329} a^{22} + \frac{1}{33826329} a^{21} - \frac{16}{33826329} a^{20} + \frac{13}{33826329} a^{19} + \frac{457}{2187} a^{18} + \frac{221}{2187} a^{17} + \frac{595}{2187} a^{16} + \frac{929}{2187} a^{15} - \frac{527}{2187} a^{14} - \frac{73}{2187} a^{13} - \frac{533}{2187} a^{12} + \frac{752}{2187} a^{11} + \frac{847}{2187} a^{10} - \frac{916}{2187} a^{9} + \frac{562}{2187} a^{8} - \frac{1}{2187} a^{7} + \frac{10064}{11275443} a^{6} + \frac{1801}{3758481} a^{5} - \frac{3955}{1252827} a^{4} + \frac{718}{417609} a^{3} + \frac{1079}{139203} a^{2} - \frac{599}{46401} a - \frac{160}{15467}$, $\frac{1}{101478987} a^{26} - \frac{1}{101478987} a^{25} - \frac{2}{101478987} a^{24} + \frac{5}{101478987} a^{23} + \frac{1}{101478987} a^{22} - \frac{16}{101478987} a^{21} + \frac{13}{101478987} a^{20} + \frac{35}{101478987} a^{19} + \frac{221}{6561} a^{18} - \frac{1592}{6561} a^{17} + \frac{929}{6561} a^{16} - \frac{2714}{6561} a^{15} - \frac{73}{6561} a^{14} + \frac{1654}{6561} a^{13} - \frac{1435}{6561} a^{12} + \frac{3034}{6561} a^{11} + \frac{1271}{6561} a^{10} + \frac{2749}{6561} a^{9} - \frac{1}{6561} a^{8} + \frac{10064}{33826329} a^{7} + \frac{1801}{11275443} a^{6} - \frac{3955}{3758481} a^{5} + \frac{718}{1252827} a^{4} + \frac{1079}{417609} a^{3} - \frac{599}{139203} a^{2} - \frac{160}{46401} a + \frac{253}{15467}$, $\frac{1}{304436961} a^{27} - \frac{1}{304436961} a^{26} - \frac{2}{304436961} a^{25} + \frac{5}{304436961} a^{24} + \frac{1}{304436961} a^{23} - \frac{16}{304436961} a^{22} + \frac{13}{304436961} a^{21} + \frac{35}{304436961} a^{20} - \frac{74}{304436961} a^{19} - \frac{1592}{19683} a^{18} + \frac{929}{19683} a^{17} + \frac{3847}{19683} a^{16} - \frac{6634}{19683} a^{15} - \frac{4907}{19683} a^{14} + \frac{5126}{19683} a^{13} + \frac{9595}{19683} a^{12} - \frac{5290}{19683} a^{11} - \frac{3812}{19683} a^{10} - \frac{1}{19683} a^{9} + \frac{10064}{101478987} a^{8} + \frac{1801}{33826329} a^{7} - \frac{3955}{11275443} a^{6} + \frac{718}{3758481} a^{5} + \frac{1079}{1252827} a^{4} - \frac{599}{417609} a^{3} - \frac{160}{139203} a^{2} + \frac{253}{46401} a - \frac{31}{15467}$, $\frac{1}{913310883} a^{28} - \frac{1}{913310883} a^{27} - \frac{2}{913310883} a^{26} + \frac{5}{913310883} a^{25} + \frac{1}{913310883} a^{24} - \frac{16}{913310883} a^{23} + \frac{13}{913310883} a^{22} + \frac{35}{913310883} a^{21} - \frac{74}{913310883} a^{20} - \frac{31}{913310883} a^{19} + \frac{20612}{59049} a^{18} - \frac{15836}{59049} a^{17} + \frac{13049}{59049} a^{16} - \frac{24590}{59049} a^{15} - \frac{14557}{59049} a^{14} + \frac{29278}{59049} a^{13} + \frac{14393}{59049} a^{12} + \frac{15871}{59049} a^{11} - \frac{1}{59049} a^{10} + \frac{10064}{304436961} a^{9} + \frac{1801}{101478987} a^{8} - \frac{3955}{33826329} a^{7} + \frac{718}{11275443} a^{6} + \frac{1079}{3758481} a^{5} - \frac{599}{1252827} a^{4} - \frac{160}{417609} a^{3} + \frac{253}{139203} a^{2} - \frac{31}{46401} a - \frac{74}{15467}$, $\frac{1}{2739932649} a^{29} - \frac{1}{2739932649} a^{28} - \frac{2}{2739932649} a^{27} + \frac{5}{2739932649} a^{26} + \frac{1}{2739932649} a^{25} - \frac{16}{2739932649} a^{24} + \frac{13}{2739932649} a^{23} + \frac{35}{2739932649} a^{22} - \frac{74}{2739932649} a^{21} - \frac{31}{2739932649} a^{20} + \frac{253}{2739932649} a^{19} + \frac{43213}{177147} a^{18} + \frac{72098}{177147} a^{17} - \frac{24590}{177147} a^{16} - \frac{14557}{177147} a^{15} + \frac{88327}{177147} a^{14} - \frac{44656}{177147} a^{13} - \frac{43178}{177147} a^{12} - \frac{1}{177147} a^{11} + \frac{10064}{913310883} a^{10} + \frac{1801}{304436961} a^{9} - \frac{3955}{101478987} a^{8} + \frac{718}{33826329} a^{7} + \frac{1079}{11275443} a^{6} - \frac{599}{3758481} a^{5} - \frac{160}{1252827} a^{4} + \frac{253}{417609} a^{3} - \frac{31}{139203} a^{2} - \frac{74}{46401} a + \frac{35}{15467}$, $\frac{1}{8219797947} a^{30} - \frac{1}{8219797947} a^{29} - \frac{2}{8219797947} a^{28} + \frac{5}{8219797947} a^{27} + \frac{1}{8219797947} a^{26} - \frac{16}{8219797947} a^{25} + \frac{13}{8219797947} a^{24} + \frac{35}{8219797947} a^{23} - \frac{74}{8219797947} a^{22} - \frac{31}{8219797947} a^{21} + \frac{253}{8219797947} a^{20} - \frac{160}{8219797947} a^{19} + \frac{249245}{531441} a^{18} + \frac{152557}{531441} a^{17} + \frac{162590}{531441} a^{16} - \frac{88820}{531441} a^{15} + \frac{132491}{531441} a^{14} + \frac{133969}{531441} a^{13} - \frac{1}{531441} a^{12} + \frac{10064}{2739932649} a^{11} + \frac{1801}{913310883} a^{10} - \frac{3955}{304436961} a^{9} + \frac{718}{101478987} a^{8} + \frac{1079}{33826329} a^{7} - \frac{599}{11275443} a^{6} - \frac{160}{3758481} a^{5} + \frac{253}{1252827} a^{4} - \frac{31}{417609} a^{3} - \frac{74}{139203} a^{2} + \frac{35}{46401} a + \frac{13}{15467}$, $\frac{1}{24659393841} a^{31} - \frac{1}{24659393841} a^{30} - \frac{2}{24659393841} a^{29} + \frac{5}{24659393841} a^{28} + \frac{1}{24659393841} a^{27} - \frac{16}{24659393841} a^{26} + \frac{13}{24659393841} a^{25} + \frac{35}{24659393841} a^{24} - \frac{74}{24659393841} a^{23} - \frac{31}{24659393841} a^{22} + \frac{253}{24659393841} a^{21} - \frac{160}{24659393841} a^{20} - \frac{599}{24659393841} a^{19} + \frac{152557}{1594323} a^{18} + \frac{694031}{1594323} a^{17} + \frac{442621}{1594323} a^{16} + \frac{663932}{1594323} a^{15} - \frac{397472}{1594323} a^{14} - \frac{1}{1594323} a^{13} + \frac{10064}{8219797947} a^{12} + \frac{1801}{2739932649} a^{11} - \frac{3955}{913310883} a^{10} + \frac{718}{304436961} a^{9} + \frac{1079}{101478987} a^{8} - \frac{599}{33826329} a^{7} - \frac{160}{11275443} a^{6} + \frac{253}{3758481} a^{5} - \frac{31}{1252827} a^{4} - \frac{74}{417609} a^{3} + \frac{35}{139203} a^{2} + \frac{13}{46401} a - \frac{16}{15467}$, $\frac{1}{73978181523} a^{32} - \frac{1}{73978181523} a^{31} - \frac{2}{73978181523} a^{30} + \frac{5}{73978181523} a^{29} + \frac{1}{73978181523} a^{28} - \frac{16}{73978181523} a^{27} + \frac{13}{73978181523} a^{26} + \frac{35}{73978181523} a^{25} - \frac{74}{73978181523} a^{24} - \frac{31}{73978181523} a^{23} + \frac{253}{73978181523} a^{22} - \frac{160}{73978181523} a^{21} - \frac{599}{73978181523} a^{20} + \frac{1079}{73978181523} a^{19} + \frac{2288354}{4782969} a^{18} + \frac{2036944}{4782969} a^{17} + \frac{663932}{4782969} a^{16} - \frac{1991795}{4782969} a^{15} - \frac{1}{4782969} a^{14} + \frac{10064}{24659393841} a^{13} + \frac{1801}{8219797947} a^{12} - \frac{3955}{2739932649} a^{11} + \frac{718}{913310883} a^{10} + \frac{1079}{304436961} a^{9} - \frac{599}{101478987} a^{8} - \frac{160}{33826329} a^{7} + \frac{253}{11275443} a^{6} - \frac{31}{3758481} a^{5} - \frac{74}{1252827} a^{4} + \frac{35}{417609} a^{3} + \frac{13}{139203} a^{2} - \frac{16}{46401} a + \frac{1}{15467}$, $\frac{1}{221934544569} a^{33} - \frac{1}{221934544569} a^{32} - \frac{2}{221934544569} a^{31} + \frac{5}{221934544569} a^{30} + \frac{1}{221934544569} a^{29} - \frac{16}{221934544569} a^{28} + \frac{13}{221934544569} a^{27} + \frac{35}{221934544569} a^{26} - \frac{74}{221934544569} a^{25} - \frac{31}{221934544569} a^{24} + \frac{253}{221934544569} a^{23} - \frac{160}{221934544569} a^{22} - \frac{599}{221934544569} a^{21} + \frac{1079}{221934544569} a^{20} + \frac{718}{221934544569} a^{19} - \frac{2746025}{14348907} a^{18} - \frac{4119037}{14348907} a^{17} - \frac{1991795}{14348907} a^{16} - \frac{1}{14348907} a^{15} + \frac{10064}{73978181523} a^{14} + \frac{1801}{24659393841} a^{13} - \frac{3955}{8219797947} a^{12} + \frac{718}{2739932649} a^{11} + \frac{1079}{913310883} a^{10} - \frac{599}{304436961} a^{9} - \frac{160}{101478987} a^{8} + \frac{253}{33826329} a^{7} - \frac{31}{11275443} a^{6} - \frac{74}{3758481} a^{5} + \frac{35}{1252827} a^{4} + \frac{13}{417609} a^{3} - \frac{16}{139203} a^{2} + \frac{1}{46401} a + \frac{5}{15467}$, $\frac{1}{665803633707} a^{34} - \frac{1}{665803633707} a^{33} - \frac{2}{665803633707} a^{32} + \frac{5}{665803633707} a^{31} + \frac{1}{665803633707} a^{30} - \frac{16}{665803633707} a^{29} + \frac{13}{665803633707} a^{28} + \frac{35}{665803633707} a^{27} - \frac{74}{665803633707} a^{26} - \frac{31}{665803633707} a^{25} + \frac{253}{665803633707} a^{24} - \frac{160}{665803633707} a^{23} - \frac{599}{665803633707} a^{22} + \frac{1079}{665803633707} a^{21} + \frac{718}{665803633707} a^{20} - \frac{3955}{665803633707} a^{19} - \frac{4119037}{43046721} a^{18} + \frac{12357112}{43046721} a^{17} - \frac{1}{43046721} a^{16} + \frac{10064}{221934544569} a^{15} + \frac{1801}{73978181523} a^{14} - \frac{3955}{24659393841} a^{13} + \frac{718}{8219797947} a^{12} + \frac{1079}{2739932649} a^{11} - \frac{599}{913310883} a^{10} - \frac{160}{304436961} a^{9} + \frac{253}{101478987} a^{8} - \frac{31}{33826329} a^{7} - \frac{74}{11275443} a^{6} + \frac{35}{3758481} a^{5} + \frac{13}{1252827} a^{4} - \frac{16}{417609} a^{3} + \frac{1}{139203} a^{2} + \frac{5}{46401} a - \frac{2}{15467}$, $\frac{1}{1997410901121} a^{35} - \frac{1}{1997410901121} a^{34} - \frac{2}{1997410901121} a^{33} + \frac{5}{1997410901121} a^{32} + \frac{1}{1997410901121} a^{31} - \frac{16}{1997410901121} a^{30} + \frac{13}{1997410901121} a^{29} + \frac{35}{1997410901121} a^{28} - \frac{74}{1997410901121} a^{27} - \frac{31}{1997410901121} a^{26} + \frac{253}{1997410901121} a^{25} - \frac{160}{1997410901121} a^{24} - \frac{599}{1997410901121} a^{23} + \frac{1079}{1997410901121} a^{22} + \frac{718}{1997410901121} a^{21} - \frac{3955}{1997410901121} a^{20} + \frac{1801}{1997410901121} a^{19} + \frac{12357112}{129140163} a^{18} - \frac{1}{129140163} a^{17} + \frac{10064}{665803633707} a^{16} + \frac{1801}{221934544569} a^{15} - \frac{3955}{73978181523} a^{14} + \frac{718}{24659393841} a^{13} + \frac{1079}{8219797947} a^{12} - \frac{599}{2739932649} a^{11} - \frac{160}{913310883} a^{10} + \frac{253}{304436961} a^{9} - \frac{31}{101478987} a^{8} - \frac{74}{33826329} a^{7} + \frac{35}{11275443} a^{6} + \frac{13}{3758481} a^{5} - \frac{16}{1252827} a^{4} + \frac{1}{417609} a^{3} + \frac{5}{139203} a^{2} - \frac{2}{46401} a - \frac{1}{15467}$

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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{10064}{1997410901121} a^{35} + \frac{10064}{1997410901121} a^{34} + \frac{20128}{1997410901121} a^{33} - \frac{50320}{1997410901121} a^{32} - \frac{10064}{1997410901121} a^{31} + \frac{161024}{1997410901121} a^{30} - \frac{130832}{1997410901121} a^{29} - \frac{352240}{1997410901121} a^{28} + \frac{744736}{1997410901121} a^{27} + \frac{311984}{1997410901121} a^{26} - \frac{2546192}{1997410901121} a^{25} + \frac{1610240}{1997410901121} a^{24} + \frac{6028336}{1997410901121} a^{23} - \frac{10859056}{1997410901121} a^{22} - \frac{7225952}{1997410901121} a^{21} + \frac{39803120}{1997410901121} a^{20} - \frac{18125264}{1997410901121} a^{19} + \frac{1801}{129140163} a^{18} + \frac{10064}{129140163} a^{17} - \frac{101284096}{665803633707} a^{16} - \frac{18125264}{221934544569} a^{15} + \frac{39803120}{73978181523} a^{14} - \frac{7225952}{24659393841} a^{13} - \frac{10859056}{8219797947} a^{12} + \frac{6028336}{2739932649} a^{11} + \frac{1610240}{913310883} a^{10} - \frac{2546192}{304436961} a^{9} + \frac{311984}{101478987} a^{8} + \frac{744736}{33826329} a^{7} - \frac{352240}{11275443} a^{6} - \frac{130832}{3758481} a^{5} + \frac{161024}{1252827} a^{4} - \frac{10064}{417609} a^{3} - \frac{50320}{139203} a^{2} + \frac{20128}{46401} a + \frac{10064}{15467} \) (order $38$)
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

$C_2\times C_{18}$ (as 36T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{209}) \), \(\Q(\sqrt{-11}) \), 3.3.361.1, \(\Q(\sqrt{-11}, \sqrt{-19})\), 6.0.2476099.1, 6.6.3295687769.1, 6.0.173457251.1, \(\Q(\zeta_{19})^+\), 12.0.10861557870736197361.1, \(\Q(\zeta_{19})\), 18.18.12922465537100419689226617716849.1, 18.0.680129765110548404696137774571.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ R $18^{2}$ $18^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19Data not computed