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Properties

Label 36.0.42237968036...6489.1
Degree $36$
Signature $[0, 18]$
Discriminant $17^{18}\cdot 19^{34}$
Root discriminant $66.52$
Ramified primes $17, 19$
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![68719476736, 17179869184, 21474836480, 9663676416, 7784628224, 4362076160, 3036676096, 1849688064, 1221591040, 767819776, 497352704, 316293120, 203411456, 129926144, 83334400, 53315136, 34162384, 21869380, 14007941, -5467345, 2135149, -833049, 325525, -126881, 49661, -19305, 7589, -2929, 1165, -441, 181, -65, 29, -9, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 5*x^34 - 9*x^33 + 29*x^32 - 65*x^31 + 181*x^30 - 441*x^29 + 1165*x^28 - 2929*x^27 + 7589*x^26 - 19305*x^25 + 49661*x^24 - 126881*x^23 + 325525*x^22 - 833049*x^21 + 2135149*x^20 - 5467345*x^19 + 14007941*x^18 + 21869380*x^17 + 34162384*x^16 + 53315136*x^15 + 83334400*x^14 + 129926144*x^13 + 203411456*x^12 + 316293120*x^11 + 497352704*x^10 + 767819776*x^9 + 1221591040*x^8 + 1849688064*x^7 + 3036676096*x^6 + 4362076160*x^5 + 7784628224*x^4 + 9663676416*x^3 + 21474836480*x^2 + 17179869184*x + 68719476736)
 
gp: K = bnfinit(x^36 - x^35 + 5*x^34 - 9*x^33 + 29*x^32 - 65*x^31 + 181*x^30 - 441*x^29 + 1165*x^28 - 2929*x^27 + 7589*x^26 - 19305*x^25 + 49661*x^24 - 126881*x^23 + 325525*x^22 - 833049*x^21 + 2135149*x^20 - 5467345*x^19 + 14007941*x^18 + 21869380*x^17 + 34162384*x^16 + 53315136*x^15 + 83334400*x^14 + 129926144*x^13 + 203411456*x^12 + 316293120*x^11 + 497352704*x^10 + 767819776*x^9 + 1221591040*x^8 + 1849688064*x^7 + 3036676096*x^6 + 4362076160*x^5 + 7784628224*x^4 + 9663676416*x^3 + 21474836480*x^2 + 17179869184*x + 68719476736, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 5 x^{34} - 9 x^{33} + 29 x^{32} - 65 x^{31} + 181 x^{30} - 441 x^{29} + 1165 x^{28} - 2929 x^{27} + 7589 x^{26} - 19305 x^{25} + 49661 x^{24} - 126881 x^{23} + 325525 x^{22} - 833049 x^{21} + 2135149 x^{20} - 5467345 x^{19} + 14007941 x^{18} + 21869380 x^{17} + 34162384 x^{16} + 53315136 x^{15} + 83334400 x^{14} + 129926144 x^{13} + 203411456 x^{12} + 316293120 x^{11} + 497352704 x^{10} + 767819776 x^{9} + 1221591040 x^{8} + 1849688064 x^{7} + 3036676096 x^{6} + 4362076160 x^{5} + 7784628224 x^{4} + 9663676416 x^{3} + 21474836480 x^{2} + 17179869184 x + 68719476736 \)

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:  \(422379680368177798715623889905033676852479341516255339435483316489=17^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.52$
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:  $17, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(323=17\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{323}(256,·)$, $\chi_{323}(1,·)$, $\chi_{323}(135,·)$, $\chi_{323}(137,·)$, $\chi_{323}(271,·)$, $\chi_{323}(16,·)$, $\chi_{323}(273,·)$, $\chi_{323}(18,·)$, $\chi_{323}(154,·)$, $\chi_{323}(288,·)$, $\chi_{323}(33,·)$, $\chi_{323}(290,·)$, $\chi_{323}(35,·)$, $\chi_{323}(169,·)$, $\chi_{323}(305,·)$, $\chi_{323}(50,·)$, $\chi_{323}(307,·)$, $\chi_{323}(52,·)$, $\chi_{323}(186,·)$, $\chi_{323}(188,·)$, $\chi_{323}(322,·)$, $\chi_{323}(67,·)$, $\chi_{323}(69,·)$, $\chi_{323}(203,·)$, $\chi_{323}(205,·)$, $\chi_{323}(84,·)$, $\chi_{323}(86,·)$, $\chi_{323}(220,·)$, $\chi_{323}(222,·)$, $\chi_{323}(101,·)$, $\chi_{323}(103,·)$, $\chi_{323}(237,·)$, $\chi_{323}(239,·)$, $\chi_{323}(118,·)$, $\chi_{323}(120,·)$, $\chi_{323}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{56031764} a^{19} - \frac{1}{4} a^{18} + \frac{1}{4} a^{17} - \frac{1}{4} a^{16} + \frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{5467345}{14007941}$, $\frac{1}{224127056} a^{20} - \frac{1}{224127056} a^{19} + \frac{1}{16} a^{18} - \frac{5}{16} a^{17} - \frac{7}{16} a^{16} + \frac{3}{16} a^{15} + \frac{1}{16} a^{14} - \frac{5}{16} a^{13} - \frac{7}{16} a^{12} + \frac{3}{16} a^{11} + \frac{1}{16} a^{10} - \frac{5}{16} a^{9} - \frac{7}{16} a^{8} + \frac{3}{16} a^{7} + \frac{1}{16} a^{6} - \frac{5}{16} a^{5} - \frac{7}{16} a^{4} + \frac{3}{16} a^{3} + \frac{1}{16} a^{2} + \frac{5467345}{56031764} a + \frac{2135149}{14007941}$, $\frac{1}{896508224} a^{21} - \frac{1}{896508224} a^{20} + \frac{5}{896508224} a^{19} + \frac{11}{64} a^{18} - \frac{7}{64} a^{17} - \frac{13}{64} a^{16} - \frac{15}{64} a^{15} + \frac{27}{64} a^{14} - \frac{23}{64} a^{13} + \frac{3}{64} a^{12} - \frac{31}{64} a^{11} - \frac{21}{64} a^{10} + \frac{25}{64} a^{9} + \frac{19}{64} a^{8} + \frac{17}{64} a^{7} - \frac{5}{64} a^{6} + \frac{9}{64} a^{5} - \frac{29}{64} a^{4} + \frac{1}{64} a^{3} + \frac{5467345}{224127056} a^{2} + \frac{2135149}{56031764} a + \frac{833049}{14007941}$, $\frac{1}{3586032896} a^{22} - \frac{1}{3586032896} a^{21} + \frac{5}{3586032896} a^{20} - \frac{9}{3586032896} a^{19} - \frac{71}{256} a^{18} + \frac{115}{256} a^{17} + \frac{113}{256} a^{16} + \frac{91}{256} a^{15} + \frac{105}{256} a^{14} + \frac{3}{256} a^{13} - \frac{95}{256} a^{12} + \frac{107}{256} a^{11} + \frac{25}{256} a^{10} - \frac{109}{256} a^{9} - \frac{47}{256} a^{8} + \frac{123}{256} a^{7} - \frac{55}{256} a^{6} + \frac{35}{256} a^{5} + \frac{1}{256} a^{4} + \frac{5467345}{896508224} a^{3} + \frac{2135149}{224127056} a^{2} + \frac{833049}{56031764} a + \frac{325525}{14007941}$, $\frac{1}{14344131584} a^{23} - \frac{1}{14344131584} a^{22} + \frac{5}{14344131584} a^{21} - \frac{9}{14344131584} a^{20} + \frac{29}{14344131584} a^{19} - \frac{397}{1024} a^{18} + \frac{113}{1024} a^{17} + \frac{347}{1024} a^{16} + \frac{105}{1024} a^{15} + \frac{259}{1024} a^{14} + \frac{161}{1024} a^{13} - \frac{149}{1024} a^{12} - \frac{231}{1024} a^{11} - \frac{365}{1024} a^{10} + \frac{465}{1024} a^{9} + \frac{123}{1024} a^{8} - \frac{311}{1024} a^{7} - \frac{221}{1024} a^{6} + \frac{1}{1024} a^{5} + \frac{5467345}{3586032896} a^{4} + \frac{2135149}{896508224} a^{3} + \frac{833049}{224127056} a^{2} + \frac{325525}{56031764} a + \frac{126881}{14007941}$, $\frac{1}{57376526336} a^{24} - \frac{1}{57376526336} a^{23} + \frac{5}{57376526336} a^{22} - \frac{9}{57376526336} a^{21} + \frac{29}{57376526336} a^{20} - \frac{65}{57376526336} a^{19} - \frac{1935}{4096} a^{18} + \frac{347}{4096} a^{17} + \frac{105}{4096} a^{16} + \frac{1283}{4096} a^{15} - \frac{863}{4096} a^{14} + \frac{1899}{4096} a^{13} - \frac{1255}{4096} a^{12} + \frac{659}{4096} a^{11} - \frac{1583}{4096} a^{10} + \frac{123}{4096} a^{9} + \frac{1737}{4096} a^{8} - \frac{1245}{4096} a^{7} + \frac{1}{4096} a^{6} + \frac{5467345}{14344131584} a^{5} + \frac{2135149}{3586032896} a^{4} + \frac{833049}{896508224} a^{3} + \frac{325525}{224127056} a^{2} + \frac{126881}{56031764} a + \frac{49661}{14007941}$, $\frac{1}{229506105344} a^{25} - \frac{1}{229506105344} a^{24} + \frac{5}{229506105344} a^{23} - \frac{9}{229506105344} a^{22} + \frac{29}{229506105344} a^{21} - \frac{65}{229506105344} a^{20} + \frac{181}{229506105344} a^{19} + \frac{347}{16384} a^{18} - \frac{8087}{16384} a^{17} - \frac{6909}{16384} a^{16} + \frac{7329}{16384} a^{15} - \frac{2197}{16384} a^{14} - \frac{1255}{16384} a^{13} - \frac{7533}{16384} a^{12} + \frac{2513}{16384} a^{11} + \frac{123}{16384} a^{10} - \frac{6455}{16384} a^{9} + \frac{6947}{16384} a^{8} + \frac{1}{16384} a^{7} + \frac{5467345}{57376526336} a^{6} + \frac{2135149}{14344131584} a^{5} + \frac{833049}{3586032896} a^{4} + \frac{325525}{896508224} a^{3} + \frac{126881}{224127056} a^{2} + \frac{49661}{56031764} a + \frac{19305}{14007941}$, $\frac{1}{918024421376} a^{26} - \frac{1}{918024421376} a^{25} + \frac{5}{918024421376} a^{24} - \frac{9}{918024421376} a^{23} + \frac{29}{918024421376} a^{22} - \frac{65}{918024421376} a^{21} + \frac{181}{918024421376} a^{20} - \frac{441}{918024421376} a^{19} + \frac{24681}{65536} a^{18} - \frac{23293}{65536} a^{17} - \frac{9055}{65536} a^{16} - \frac{18581}{65536} a^{15} - \frac{17639}{65536} a^{14} + \frac{8851}{65536} a^{13} - \frac{13871}{65536} a^{12} - \frac{16261}{65536} a^{11} + \frac{26313}{65536} a^{10} - \frac{25821}{65536} a^{9} + \frac{1}{65536} a^{8} + \frac{5467345}{229506105344} a^{7} + \frac{2135149}{57376526336} a^{6} + \frac{833049}{14344131584} a^{5} + \frac{325525}{3586032896} a^{4} + \frac{126881}{896508224} a^{3} + \frac{49661}{224127056} a^{2} + \frac{19305}{56031764} a + \frac{7589}{14007941}$, $\frac{1}{3672097685504} a^{27} - \frac{1}{3672097685504} a^{26} + \frac{5}{3672097685504} a^{25} - \frac{9}{3672097685504} a^{24} + \frac{29}{3672097685504} a^{23} - \frac{65}{3672097685504} a^{22} + \frac{181}{3672097685504} a^{21} - \frac{441}{3672097685504} a^{20} + \frac{1165}{3672097685504} a^{19} - \frac{23293}{262144} a^{18} + \frac{122017}{262144} a^{17} + \frac{46955}{262144} a^{16} - \frac{83175}{262144} a^{15} + \frac{8851}{262144} a^{14} - \frac{79407}{262144} a^{13} + \frac{114811}{262144} a^{12} + \frac{91849}{262144} a^{11} + \frac{105251}{262144} a^{10} + \frac{1}{262144} a^{9} + \frac{5467345}{918024421376} a^{8} + \frac{2135149}{229506105344} a^{7} + \frac{833049}{57376526336} a^{6} + \frac{325525}{14344131584} a^{5} + \frac{126881}{3586032896} a^{4} + \frac{49661}{896508224} a^{3} + \frac{19305}{224127056} a^{2} + \frac{7589}{56031764} a + \frac{2929}{14007941}$, $\frac{1}{14688390742016} a^{28} - \frac{1}{14688390742016} a^{27} + \frac{5}{14688390742016} a^{26} - \frac{9}{14688390742016} a^{25} + \frac{29}{14688390742016} a^{24} - \frac{65}{14688390742016} a^{23} + \frac{181}{14688390742016} a^{22} - \frac{441}{14688390742016} a^{21} + \frac{1165}{14688390742016} a^{20} - \frac{2929}{14688390742016} a^{19} + \frac{384161}{1048576} a^{18} - \frac{477333}{1048576} a^{17} - \frac{83175}{1048576} a^{16} + \frac{270995}{1048576} a^{15} + \frac{444881}{1048576} a^{14} - \frac{409477}{1048576} a^{13} + \frac{91849}{1048576} a^{12} + \frac{367395}{1048576} a^{11} + \frac{1}{1048576} a^{10} + \frac{5467345}{3672097685504} a^{9} + \frac{2135149}{918024421376} a^{8} + \frac{833049}{229506105344} a^{7} + \frac{325525}{57376526336} a^{6} + \frac{126881}{14344131584} a^{5} + \frac{49661}{3586032896} a^{4} + \frac{19305}{896508224} a^{3} + \frac{7589}{224127056} a^{2} + \frac{2929}{56031764} a + \frac{1165}{14007941}$, $\frac{1}{58753562968064} a^{29} - \frac{1}{58753562968064} a^{28} + \frac{5}{58753562968064} a^{27} - \frac{9}{58753562968064} a^{26} + \frac{29}{58753562968064} a^{25} - \frac{65}{58753562968064} a^{24} + \frac{181}{58753562968064} a^{23} - \frac{441}{58753562968064} a^{22} + \frac{1165}{58753562968064} a^{21} - \frac{2929}{58753562968064} a^{20} + \frac{7589}{58753562968064} a^{19} + \frac{1619819}{4194304} a^{18} - \frac{83175}{4194304} a^{17} - \frac{1826157}{4194304} a^{16} + \frac{1493457}{4194304} a^{15} - \frac{409477}{4194304} a^{14} - \frac{2005303}{4194304} a^{13} + \frac{367395}{4194304} a^{12} + \frac{1}{4194304} a^{11} + \frac{5467345}{14688390742016} a^{10} + \frac{2135149}{3672097685504} a^{9} + \frac{833049}{918024421376} a^{8} + \frac{325525}{229506105344} a^{7} + \frac{126881}{57376526336} a^{6} + \frac{49661}{14344131584} a^{5} + \frac{19305}{3586032896} a^{4} + \frac{7589}{896508224} a^{3} + \frac{2929}{224127056} a^{2} + \frac{1165}{56031764} a + \frac{441}{14007941}$, $\frac{1}{235014251872256} a^{30} - \frac{1}{235014251872256} a^{29} + \frac{5}{235014251872256} a^{28} - \frac{9}{235014251872256} a^{27} + \frac{29}{235014251872256} a^{26} - \frac{65}{235014251872256} a^{25} + \frac{181}{235014251872256} a^{24} - \frac{441}{235014251872256} a^{23} + \frac{1165}{235014251872256} a^{22} - \frac{2929}{235014251872256} a^{21} + \frac{7589}{235014251872256} a^{20} - \frac{19305}{235014251872256} a^{19} - \frac{83175}{16777216} a^{18} + \frac{6562451}{16777216} a^{17} - \frac{6895151}{16777216} a^{16} - \frac{409477}{16777216} a^{15} + \frac{6383305}{16777216} a^{14} - \frac{8021213}{16777216} a^{13} + \frac{1}{16777216} a^{12} + \frac{5467345}{58753562968064} a^{11} + \frac{2135149}{14688390742016} a^{10} + \frac{833049}{3672097685504} a^{9} + \frac{325525}{918024421376} a^{8} + \frac{126881}{229506105344} a^{7} + \frac{49661}{57376526336} a^{6} + \frac{19305}{14344131584} a^{5} + \frac{7589}{3586032896} a^{4} + \frac{2929}{896508224} a^{3} + \frac{1165}{224127056} a^{2} + \frac{441}{56031764} a + \frac{181}{14007941}$, $\frac{1}{940057007489024} a^{31} - \frac{1}{940057007489024} a^{30} + \frac{5}{940057007489024} a^{29} - \frac{9}{940057007489024} a^{28} + \frac{29}{940057007489024} a^{27} - \frac{65}{940057007489024} a^{26} + \frac{181}{940057007489024} a^{25} - \frac{441}{940057007489024} a^{24} + \frac{1165}{940057007489024} a^{23} - \frac{2929}{940057007489024} a^{22} + \frac{7589}{940057007489024} a^{21} - \frac{19305}{940057007489024} a^{20} + \frac{49661}{940057007489024} a^{19} - \frac{26991981}{67108864} a^{18} + \frac{26659281}{67108864} a^{17} - \frac{409477}{67108864} a^{16} - \frac{27171127}{67108864} a^{15} + \frac{25533219}{67108864} a^{14} + \frac{1}{67108864} a^{13} + \frac{5467345}{235014251872256} a^{12} + \frac{2135149}{58753562968064} a^{11} + \frac{833049}{14688390742016} a^{10} + \frac{325525}{3672097685504} a^{9} + \frac{126881}{918024421376} a^{8} + \frac{49661}{229506105344} a^{7} + \frac{19305}{57376526336} a^{6} + \frac{7589}{14344131584} a^{5} + \frac{2929}{3586032896} a^{4} + \frac{1165}{896508224} a^{3} + \frac{441}{224127056} a^{2} + \frac{181}{56031764} a + \frac{65}{14007941}$, $\frac{1}{3760228029956096} a^{32} - \frac{1}{3760228029956096} a^{31} + \frac{5}{3760228029956096} a^{30} - \frac{9}{3760228029956096} a^{29} + \frac{29}{3760228029956096} a^{28} - \frac{65}{3760228029956096} a^{27} + \frac{181}{3760228029956096} a^{26} - \frac{441}{3760228029956096} a^{25} + \frac{1165}{3760228029956096} a^{24} - \frac{2929}{3760228029956096} a^{23} + \frac{7589}{3760228029956096} a^{22} - \frac{19305}{3760228029956096} a^{21} + \frac{49661}{3760228029956096} a^{20} - \frac{126881}{3760228029956096} a^{19} - \frac{107558447}{268435456} a^{18} - \frac{409477}{268435456} a^{17} + \frac{107046601}{268435456} a^{16} - \frac{108684509}{268435456} a^{15} + \frac{1}{268435456} a^{14} + \frac{5467345}{940057007489024} a^{13} + \frac{2135149}{235014251872256} a^{12} + \frac{833049}{58753562968064} a^{11} + \frac{325525}{14688390742016} a^{10} + \frac{126881}{3672097685504} a^{9} + \frac{49661}{918024421376} a^{8} + \frac{19305}{229506105344} a^{7} + \frac{7589}{57376526336} a^{6} + \frac{2929}{14344131584} a^{5} + \frac{1165}{3586032896} a^{4} + \frac{441}{896508224} a^{3} + \frac{181}{224127056} a^{2} + \frac{65}{56031764} a + \frac{29}{14007941}$, $\frac{1}{15040912119824384} a^{33} - \frac{1}{15040912119824384} a^{32} + \frac{5}{15040912119824384} a^{31} - \frac{9}{15040912119824384} a^{30} + \frac{29}{15040912119824384} a^{29} - \frac{65}{15040912119824384} a^{28} + \frac{181}{15040912119824384} a^{27} - \frac{441}{15040912119824384} a^{26} + \frac{1165}{15040912119824384} a^{25} - \frac{2929}{15040912119824384} a^{24} + \frac{7589}{15040912119824384} a^{23} - \frac{19305}{15040912119824384} a^{22} + \frac{49661}{15040912119824384} a^{21} - \frac{126881}{15040912119824384} a^{20} + \frac{325525}{15040912119824384} a^{19} - \frac{409477}{1073741824} a^{18} - \frac{429824311}{1073741824} a^{17} + \frac{428186403}{1073741824} a^{16} + \frac{1}{1073741824} a^{15} + \frac{5467345}{3760228029956096} a^{14} + \frac{2135149}{940057007489024} a^{13} + \frac{833049}{235014251872256} a^{12} + \frac{325525}{58753562968064} a^{11} + \frac{126881}{14688390742016} a^{10} + \frac{49661}{3672097685504} a^{9} + \frac{19305}{918024421376} a^{8} + \frac{7589}{229506105344} a^{7} + \frac{2929}{57376526336} a^{6} + \frac{1165}{14344131584} a^{5} + \frac{441}{3586032896} a^{4} + \frac{181}{896508224} a^{3} + \frac{65}{224127056} a^{2} + \frac{29}{56031764} a + \frac{9}{14007941}$, $\frac{1}{60163648479297536} a^{34} - \frac{1}{60163648479297536} a^{33} + \frac{5}{60163648479297536} a^{32} - \frac{9}{60163648479297536} a^{31} + \frac{29}{60163648479297536} a^{30} - \frac{65}{60163648479297536} a^{29} + \frac{181}{60163648479297536} a^{28} - \frac{441}{60163648479297536} a^{27} + \frac{1165}{60163648479297536} a^{26} - \frac{2929}{60163648479297536} a^{25} + \frac{7589}{60163648479297536} a^{24} - \frac{19305}{60163648479297536} a^{23} + \frac{49661}{60163648479297536} a^{22} - \frac{126881}{60163648479297536} a^{21} + \frac{325525}{60163648479297536} a^{20} - \frac{833049}{60163648479297536} a^{19} + \frac{1717659337}{4294967296} a^{18} - \frac{1719297245}{4294967296} a^{17} + \frac{1}{4294967296} a^{16} + \frac{5467345}{15040912119824384} a^{15} + \frac{2135149}{3760228029956096} a^{14} + \frac{833049}{940057007489024} a^{13} + \frac{325525}{235014251872256} a^{12} + \frac{126881}{58753562968064} a^{11} + \frac{49661}{14688390742016} a^{10} + \frac{19305}{3672097685504} a^{9} + \frac{7589}{918024421376} a^{8} + \frac{2929}{229506105344} a^{7} + \frac{1165}{57376526336} a^{6} + \frac{441}{14344131584} a^{5} + \frac{181}{3586032896} a^{4} + \frac{65}{896508224} a^{3} + \frac{29}{224127056} a^{2} + \frac{9}{56031764} a + \frac{5}{14007941}$, $\frac{1}{240654593917190144} a^{35} - \frac{1}{240654593917190144} a^{34} + \frac{5}{240654593917190144} a^{33} - \frac{9}{240654593917190144} a^{32} + \frac{29}{240654593917190144} a^{31} - \frac{65}{240654593917190144} a^{30} + \frac{181}{240654593917190144} a^{29} - \frac{441}{240654593917190144} a^{28} + \frac{1165}{240654593917190144} a^{27} - \frac{2929}{240654593917190144} a^{26} + \frac{7589}{240654593917190144} a^{25} - \frac{19305}{240654593917190144} a^{24} + \frac{49661}{240654593917190144} a^{23} - \frac{126881}{240654593917190144} a^{22} + \frac{325525}{240654593917190144} a^{21} - \frac{833049}{240654593917190144} a^{20} + \frac{2135149}{240654593917190144} a^{19} + \frac{6870637347}{17179869184} a^{18} + \frac{1}{17179869184} a^{17} + \frac{5467345}{60163648479297536} a^{16} + \frac{2135149}{15040912119824384} a^{15} + \frac{833049}{3760228029956096} a^{14} + \frac{325525}{940057007489024} a^{13} + \frac{126881}{235014251872256} a^{12} + \frac{49661}{58753562968064} a^{11} + \frac{19305}{14688390742016} a^{10} + \frac{7589}{3672097685504} a^{9} + \frac{2929}{918024421376} a^{8} + \frac{1165}{229506105344} a^{7} + \frac{441}{57376526336} a^{6} + \frac{181}{14344131584} a^{5} + \frac{65}{3586032896} a^{4} + \frac{29}{896508224} a^{3} + \frac{9}{224127056} a^{2} + \frac{5}{56031764} a + \frac{1}{14007941}$

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:  \( -\frac{1165}{3672097685504} a^{28} - \frac{66507086889}{3672097685504} a^{9} \) (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

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-323}) \), 3.3.361.1, \(\Q(\sqrt{17}, \sqrt{-19})\), 6.0.2476099.1, 6.6.640267073.1, 6.0.12165074387.1, \(\Q(\zeta_{19})^+\), 12.0.147989034841243425769.1, \(\Q(\zeta_{19})\), 18.18.34205654728777159191037355893457.1, 18.0.649907439846766024629709761975683.1

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 $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ R R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$

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
17Data not computed
19Data not computed