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Properties

Label 36.0.48908816365...2729.1
Degree $36$
Signature $[0, 18]$
Discriminant $7^{18}\cdot 19^{34}$
Root discriminant $42.68$
Ramified primes $7, 19$
Class number $148$ (GRH)
Class group $[2, 74]$ (GRH)
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![262144, -131072, -65536, 98304, -16384, -40960, 28672, 6144, -17408, 5632, 5888, -5760, -64, 2912, -1424, -744, 1084, -170, -457, -85, 271, -93, -89, 91, -1, -45, 23, 11, -17, 3, 7, -5, -1, 3, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - x^34 + 3*x^33 - x^32 - 5*x^31 + 7*x^30 + 3*x^29 - 17*x^28 + 11*x^27 + 23*x^26 - 45*x^25 - x^24 + 91*x^23 - 89*x^22 - 93*x^21 + 271*x^20 - 85*x^19 - 457*x^18 - 170*x^17 + 1084*x^16 - 744*x^15 - 1424*x^14 + 2912*x^13 - 64*x^12 - 5760*x^11 + 5888*x^10 + 5632*x^9 - 17408*x^8 + 6144*x^7 + 28672*x^6 - 40960*x^5 - 16384*x^4 + 98304*x^3 - 65536*x^2 - 131072*x + 262144)
 
gp: K = bnfinit(x^36 - x^35 - x^34 + 3*x^33 - x^32 - 5*x^31 + 7*x^30 + 3*x^29 - 17*x^28 + 11*x^27 + 23*x^26 - 45*x^25 - x^24 + 91*x^23 - 89*x^22 - 93*x^21 + 271*x^20 - 85*x^19 - 457*x^18 - 170*x^17 + 1084*x^16 - 744*x^15 - 1424*x^14 + 2912*x^13 - 64*x^12 - 5760*x^11 + 5888*x^10 + 5632*x^9 - 17408*x^8 + 6144*x^7 + 28672*x^6 - 40960*x^5 - 16384*x^4 + 98304*x^3 - 65536*x^2 - 131072*x + 262144, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - x^{34} + 3 x^{33} - x^{32} - 5 x^{31} + 7 x^{30} + 3 x^{29} - 17 x^{28} + 11 x^{27} + 23 x^{26} - 45 x^{25} - x^{24} + 91 x^{23} - 89 x^{22} - 93 x^{21} + 271 x^{20} - 85 x^{19} - 457 x^{18} - 170 x^{17} + 1084 x^{16} - 744 x^{15} - 1424 x^{14} + 2912 x^{13} - 64 x^{12} - 5760 x^{11} + 5888 x^{10} + 5632 x^{9} - 17408 x^{8} + 6144 x^{7} + 28672 x^{6} - 40960 x^{5} - 16384 x^{4} + 98304 x^{3} - 65536 x^{2} - 131072 x + 262144 \)

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:  \(48908816365067043970916287981601635325839249495639564072729=7^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.68$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(133=7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{133}(1,·)$, $\chi_{133}(132,·)$, $\chi_{133}(6,·)$, $\chi_{133}(8,·)$, $\chi_{133}(13,·)$, $\chi_{133}(15,·)$, $\chi_{133}(20,·)$, $\chi_{133}(22,·)$, $\chi_{133}(27,·)$, $\chi_{133}(29,·)$, $\chi_{133}(34,·)$, $\chi_{133}(36,·)$, $\chi_{133}(41,·)$, $\chi_{133}(43,·)$, $\chi_{133}(48,·)$, $\chi_{133}(50,·)$, $\chi_{133}(55,·)$, $\chi_{133}(62,·)$, $\chi_{133}(64,·)$, $\chi_{133}(69,·)$, $\chi_{133}(71,·)$, $\chi_{133}(78,·)$, $\chi_{133}(83,·)$, $\chi_{133}(85,·)$, $\chi_{133}(90,·)$, $\chi_{133}(92,·)$, $\chi_{133}(97,·)$, $\chi_{133}(99,·)$, $\chi_{133}(104,·)$, $\chi_{133}(106,·)$, $\chi_{133}(111,·)$, $\chi_{133}(113,·)$, $\chi_{133}(118,·)$, $\chi_{133}(120,·)$, $\chi_{133}(125,·)$, $\chi_{133}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{914} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{85}{457}$, $\frac{1}{1828} a^{20} - \frac{1}{1828} a^{19} - \frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{85}{914} a - \frac{186}{457}$, $\frac{1}{3656} a^{21} - \frac{1}{3656} a^{20} - \frac{1}{3656} a^{19} + \frac{3}{8} a^{18} - \frac{1}{8} a^{17} + \frac{3}{8} a^{16} - \frac{1}{8} a^{15} + \frac{3}{8} a^{14} - \frac{1}{8} a^{13} + \frac{3}{8} a^{12} - \frac{1}{8} a^{11} + \frac{3}{8} a^{10} - \frac{1}{8} a^{9} + \frac{3}{8} a^{8} - \frac{1}{8} a^{7} + \frac{3}{8} a^{6} - \frac{1}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{85}{1828} a^{2} + \frac{271}{914} a - \frac{93}{457}$, $\frac{1}{7312} a^{22} - \frac{1}{7312} a^{21} - \frac{1}{7312} a^{20} + \frac{3}{7312} a^{19} + \frac{7}{16} a^{18} + \frac{3}{16} a^{17} - \frac{1}{16} a^{16} - \frac{5}{16} a^{15} + \frac{7}{16} a^{14} + \frac{3}{16} a^{13} - \frac{1}{16} a^{12} - \frac{5}{16} a^{11} + \frac{7}{16} a^{10} + \frac{3}{16} a^{9} - \frac{1}{16} a^{8} - \frac{5}{16} a^{7} + \frac{7}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{16} a^{4} - \frac{85}{3656} a^{3} + \frac{271}{1828} a^{2} - \frac{93}{914} a - \frac{89}{457}$, $\frac{1}{14624} a^{23} - \frac{1}{14624} a^{22} - \frac{1}{14624} a^{21} + \frac{3}{14624} a^{20} - \frac{1}{14624} a^{19} + \frac{3}{32} a^{18} + \frac{15}{32} a^{17} + \frac{11}{32} a^{16} - \frac{9}{32} a^{15} - \frac{13}{32} a^{14} - \frac{1}{32} a^{13} - \frac{5}{32} a^{12} + \frac{7}{32} a^{11} + \frac{3}{32} a^{10} + \frac{15}{32} a^{9} + \frac{11}{32} a^{8} - \frac{9}{32} a^{7} - \frac{13}{32} a^{6} - \frac{1}{32} a^{5} - \frac{85}{7312} a^{4} + \frac{271}{3656} a^{3} - \frac{93}{1828} a^{2} - \frac{89}{914} a + \frac{91}{457}$, $\frac{1}{29248} a^{24} - \frac{1}{29248} a^{23} - \frac{1}{29248} a^{22} + \frac{3}{29248} a^{21} - \frac{1}{29248} a^{20} - \frac{5}{29248} a^{19} + \frac{15}{64} a^{18} - \frac{21}{64} a^{17} - \frac{9}{64} a^{16} - \frac{13}{64} a^{15} + \frac{31}{64} a^{14} - \frac{5}{64} a^{13} + \frac{7}{64} a^{12} + \frac{3}{64} a^{11} - \frac{17}{64} a^{10} + \frac{11}{64} a^{9} + \frac{23}{64} a^{8} + \frac{19}{64} a^{7} - \frac{1}{64} a^{6} - \frac{85}{14624} a^{5} + \frac{271}{7312} a^{4} - \frac{93}{3656} a^{3} - \frac{89}{1828} a^{2} + \frac{91}{914} a - \frac{1}{457}$, $\frac{1}{58496} a^{25} - \frac{1}{58496} a^{24} - \frac{1}{58496} a^{23} + \frac{3}{58496} a^{22} - \frac{1}{58496} a^{21} - \frac{5}{58496} a^{20} + \frac{7}{58496} a^{19} - \frac{21}{128} a^{18} - \frac{9}{128} a^{17} + \frac{51}{128} a^{16} - \frac{33}{128} a^{15} + \frac{59}{128} a^{14} + \frac{7}{128} a^{13} + \frac{3}{128} a^{12} - \frac{17}{128} a^{11} + \frac{11}{128} a^{10} + \frac{23}{128} a^{9} - \frac{45}{128} a^{8} - \frac{1}{128} a^{7} - \frac{85}{29248} a^{6} + \frac{271}{14624} a^{5} - \frac{93}{7312} a^{4} - \frac{89}{3656} a^{3} + \frac{91}{1828} a^{2} - \frac{1}{914} a - \frac{45}{457}$, $\frac{1}{116992} a^{26} - \frac{1}{116992} a^{25} - \frac{1}{116992} a^{24} + \frac{3}{116992} a^{23} - \frac{1}{116992} a^{22} - \frac{5}{116992} a^{21} + \frac{7}{116992} a^{20} + \frac{3}{116992} a^{19} - \frac{9}{256} a^{18} + \frac{51}{256} a^{17} - \frac{33}{256} a^{16} - \frac{69}{256} a^{15} - \frac{121}{256} a^{14} + \frac{3}{256} a^{13} - \frac{17}{256} a^{12} + \frac{11}{256} a^{11} + \frac{23}{256} a^{10} - \frac{45}{256} a^{9} - \frac{1}{256} a^{8} - \frac{85}{58496} a^{7} + \frac{271}{29248} a^{6} - \frac{93}{14624} a^{5} - \frac{89}{7312} a^{4} + \frac{91}{3656} a^{3} - \frac{1}{1828} a^{2} - \frac{45}{914} a + \frac{23}{457}$, $\frac{1}{233984} a^{27} - \frac{1}{233984} a^{26} - \frac{1}{233984} a^{25} + \frac{3}{233984} a^{24} - \frac{1}{233984} a^{23} - \frac{5}{233984} a^{22} + \frac{7}{233984} a^{21} + \frac{3}{233984} a^{20} - \frac{17}{233984} a^{19} - \frac{205}{512} a^{18} + \frac{223}{512} a^{17} + \frac{187}{512} a^{16} - \frac{121}{512} a^{15} - \frac{253}{512} a^{14} - \frac{17}{512} a^{13} + \frac{11}{512} a^{12} + \frac{23}{512} a^{11} - \frac{45}{512} a^{10} - \frac{1}{512} a^{9} - \frac{85}{116992} a^{8} + \frac{271}{58496} a^{7} - \frac{93}{29248} a^{6} - \frac{89}{14624} a^{5} + \frac{91}{7312} a^{4} - \frac{1}{3656} a^{3} - \frac{45}{1828} a^{2} + \frac{23}{914} a + \frac{11}{457}$, $\frac{1}{467968} a^{28} - \frac{1}{467968} a^{27} - \frac{1}{467968} a^{26} + \frac{3}{467968} a^{25} - \frac{1}{467968} a^{24} - \frac{5}{467968} a^{23} + \frac{7}{467968} a^{22} + \frac{3}{467968} a^{21} - \frac{17}{467968} a^{20} + \frac{11}{467968} a^{19} - \frac{289}{1024} a^{18} - \frac{325}{1024} a^{17} - \frac{121}{1024} a^{16} - \frac{253}{1024} a^{15} + \frac{495}{1024} a^{14} + \frac{11}{1024} a^{13} + \frac{23}{1024} a^{12} - \frac{45}{1024} a^{11} - \frac{1}{1024} a^{10} - \frac{85}{233984} a^{9} + \frac{271}{116992} a^{8} - \frac{93}{58496} a^{7} - \frac{89}{29248} a^{6} + \frac{91}{14624} a^{5} - \frac{1}{7312} a^{4} - \frac{45}{3656} a^{3} + \frac{23}{1828} a^{2} + \frac{11}{914} a - \frac{17}{457}$, $\frac{1}{935936} a^{29} - \frac{1}{935936} a^{28} - \frac{1}{935936} a^{27} + \frac{3}{935936} a^{26} - \frac{1}{935936} a^{25} - \frac{5}{935936} a^{24} + \frac{7}{935936} a^{23} + \frac{3}{935936} a^{22} - \frac{17}{935936} a^{21} + \frac{11}{935936} a^{20} + \frac{23}{935936} a^{19} + \frac{699}{2048} a^{18} - \frac{121}{2048} a^{17} + \frac{771}{2048} a^{16} - \frac{529}{2048} a^{15} - \frac{1013}{2048} a^{14} + \frac{23}{2048} a^{13} - \frac{45}{2048} a^{12} - \frac{1}{2048} a^{11} - \frac{85}{467968} a^{10} + \frac{271}{233984} a^{9} - \frac{93}{116992} a^{8} - \frac{89}{58496} a^{7} + \frac{91}{29248} a^{6} - \frac{1}{14624} a^{5} - \frac{45}{7312} a^{4} + \frac{23}{3656} a^{3} + \frac{11}{1828} a^{2} - \frac{17}{914} a + \frac{3}{457}$, $\frac{1}{1871872} a^{30} - \frac{1}{1871872} a^{29} - \frac{1}{1871872} a^{28} + \frac{3}{1871872} a^{27} - \frac{1}{1871872} a^{26} - \frac{5}{1871872} a^{25} + \frac{7}{1871872} a^{24} + \frac{3}{1871872} a^{23} - \frac{17}{1871872} a^{22} + \frac{11}{1871872} a^{21} + \frac{23}{1871872} a^{20} - \frac{45}{1871872} a^{19} + \frac{1927}{4096} a^{18} + \frac{771}{4096} a^{17} - \frac{529}{4096} a^{16} - \frac{1013}{4096} a^{15} - \frac{2025}{4096} a^{14} - \frac{45}{4096} a^{13} - \frac{1}{4096} a^{12} - \frac{85}{935936} a^{11} + \frac{271}{467968} a^{10} - \frac{93}{233984} a^{9} - \frac{89}{116992} a^{8} + \frac{91}{58496} a^{7} - \frac{1}{29248} a^{6} - \frac{45}{14624} a^{5} + \frac{23}{7312} a^{4} + \frac{11}{3656} a^{3} - \frac{17}{1828} a^{2} + \frac{3}{914} a + \frac{7}{457}$, $\frac{1}{3743744} a^{31} - \frac{1}{3743744} a^{30} - \frac{1}{3743744} a^{29} + \frac{3}{3743744} a^{28} - \frac{1}{3743744} a^{27} - \frac{5}{3743744} a^{26} + \frac{7}{3743744} a^{25} + \frac{3}{3743744} a^{24} - \frac{17}{3743744} a^{23} + \frac{11}{3743744} a^{22} + \frac{23}{3743744} a^{21} - \frac{45}{3743744} a^{20} - \frac{1}{3743744} a^{19} + \frac{771}{8192} a^{18} + \frac{3567}{8192} a^{17} + \frac{3083}{8192} a^{16} - \frac{2025}{8192} a^{15} + \frac{4051}{8192} a^{14} - \frac{1}{8192} a^{13} - \frac{85}{1871872} a^{12} + \frac{271}{935936} a^{11} - \frac{93}{467968} a^{10} - \frac{89}{233984} a^{9} + \frac{91}{116992} a^{8} - \frac{1}{58496} a^{7} - \frac{45}{29248} a^{6} + \frac{23}{14624} a^{5} + \frac{11}{7312} a^{4} - \frac{17}{3656} a^{3} + \frac{3}{1828} a^{2} + \frac{7}{914} a - \frac{5}{457}$, $\frac{1}{7487488} a^{32} - \frac{1}{7487488} a^{31} - \frac{1}{7487488} a^{30} + \frac{3}{7487488} a^{29} - \frac{1}{7487488} a^{28} - \frac{5}{7487488} a^{27} + \frac{7}{7487488} a^{26} + \frac{3}{7487488} a^{25} - \frac{17}{7487488} a^{24} + \frac{11}{7487488} a^{23} + \frac{23}{7487488} a^{22} - \frac{45}{7487488} a^{21} - \frac{1}{7487488} a^{20} + \frac{91}{7487488} a^{19} - \frac{4625}{16384} a^{18} + \frac{3083}{16384} a^{17} + \frac{6167}{16384} a^{16} + \frac{4051}{16384} a^{15} - \frac{1}{16384} a^{14} - \frac{85}{3743744} a^{13} + \frac{271}{1871872} a^{12} - \frac{93}{935936} a^{11} - \frac{89}{467968} a^{10} + \frac{91}{233984} a^{9} - \frac{1}{116992} a^{8} - \frac{45}{58496} a^{7} + \frac{23}{29248} a^{6} + \frac{11}{14624} a^{5} - \frac{17}{7312} a^{4} + \frac{3}{3656} a^{3} + \frac{7}{1828} a^{2} - \frac{5}{914} a - \frac{1}{457}$, $\frac{1}{14974976} a^{33} - \frac{1}{14974976} a^{32} - \frac{1}{14974976} a^{31} + \frac{3}{14974976} a^{30} - \frac{1}{14974976} a^{29} - \frac{5}{14974976} a^{28} + \frac{7}{14974976} a^{27} + \frac{3}{14974976} a^{26} - \frac{17}{14974976} a^{25} + \frac{11}{14974976} a^{24} + \frac{23}{14974976} a^{23} - \frac{45}{14974976} a^{22} - \frac{1}{14974976} a^{21} + \frac{91}{14974976} a^{20} - \frac{89}{14974976} a^{19} + \frac{3083}{32768} a^{18} + \frac{6167}{32768} a^{17} - \frac{12333}{32768} a^{16} - \frac{1}{32768} a^{15} - \frac{85}{7487488} a^{14} + \frac{271}{3743744} a^{13} - \frac{93}{1871872} a^{12} - \frac{89}{935936} a^{11} + \frac{91}{467968} a^{10} - \frac{1}{233984} a^{9} - \frac{45}{116992} a^{8} + \frac{23}{58496} a^{7} + \frac{11}{29248} a^{6} - \frac{17}{14624} a^{5} + \frac{3}{7312} a^{4} + \frac{7}{3656} a^{3} - \frac{5}{1828} a^{2} - \frac{1}{914} a + \frac{3}{457}$, $\frac{1}{29949952} a^{34} - \frac{1}{29949952} a^{33} - \frac{1}{29949952} a^{32} + \frac{3}{29949952} a^{31} - \frac{1}{29949952} a^{30} - \frac{5}{29949952} a^{29} + \frac{7}{29949952} a^{28} + \frac{3}{29949952} a^{27} - \frac{17}{29949952} a^{26} + \frac{11}{29949952} a^{25} + \frac{23}{29949952} a^{24} - \frac{45}{29949952} a^{23} - \frac{1}{29949952} a^{22} + \frac{91}{29949952} a^{21} - \frac{89}{29949952} a^{20} - \frac{93}{29949952} a^{19} + \frac{6167}{65536} a^{18} - \frac{12333}{65536} a^{17} - \frac{1}{65536} a^{16} - \frac{85}{14974976} a^{15} + \frac{271}{7487488} a^{14} - \frac{93}{3743744} a^{13} - \frac{89}{1871872} a^{12} + \frac{91}{935936} a^{11} - \frac{1}{467968} a^{10} - \frac{45}{233984} a^{9} + \frac{23}{116992} a^{8} + \frac{11}{58496} a^{7} - \frac{17}{29248} a^{6} + \frac{3}{14624} a^{5} + \frac{7}{7312} a^{4} - \frac{5}{3656} a^{3} - \frac{1}{1828} a^{2} + \frac{3}{914} a - \frac{1}{457}$, $\frac{1}{59899904} a^{35} - \frac{1}{59899904} a^{34} - \frac{1}{59899904} a^{33} + \frac{3}{59899904} a^{32} - \frac{1}{59899904} a^{31} - \frac{5}{59899904} a^{30} + \frac{7}{59899904} a^{29} + \frac{3}{59899904} a^{28} - \frac{17}{59899904} a^{27} + \frac{11}{59899904} a^{26} + \frac{23}{59899904} a^{25} - \frac{45}{59899904} a^{24} - \frac{1}{59899904} a^{23} + \frac{91}{59899904} a^{22} - \frac{89}{59899904} a^{21} - \frac{93}{59899904} a^{20} + \frac{271}{59899904} a^{19} - \frac{12333}{131072} a^{18} - \frac{1}{131072} a^{17} - \frac{85}{29949952} a^{16} + \frac{271}{14974976} a^{15} - \frac{93}{7487488} a^{14} - \frac{89}{3743744} a^{13} + \frac{91}{1871872} a^{12} - \frac{1}{935936} a^{11} - \frac{45}{467968} a^{10} + \frac{23}{233984} a^{9} + \frac{11}{116992} a^{8} - \frac{17}{58496} a^{7} + \frac{3}{29248} a^{6} + \frac{7}{14624} a^{5} - \frac{5}{7312} a^{4} - \frac{1}{3656} a^{3} + \frac{3}{1828} a^{2} - \frac{1}{914} a - \frac{1}{457}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{74}$, which has order $148$ (assuming GRH)

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:  \( -\frac{3}{7312} a^{23} + \frac{967}{7312} a^{4} \) (order $38$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28076820524481.113 \) (assuming GRH)
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

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{133}) \), \(\Q(\sqrt{-7}) \), 3.3.361.1, \(\Q(\sqrt{-7}, \sqrt{-19})\), 6.0.2476099.1, 6.6.849301957.1, 6.0.44700103.1, \(\Q(\zeta_{19})^+\), 12.0.721313814164029849.1, \(\Q(\zeta_{19})\), 18.18.221153377467012797984123331973.1, 18.0.11639651445632252525480175367.1

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 $18^{2}$ $18^{2}$ $18^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ $18^{2}$ $18^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$ $18^{2}$

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
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19Data not computed