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Properties

Label 36.0.44387950739...5625.3
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 5^{18}\cdot 19^{34}$
Root discriminant $62.48$
Ramified primes $3, 5, 19$
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, -12884901888, 7516192768, 1342177280, -2214592512, 218103808, 499122176, -179306496, -79953920, 64815104, 3784704, -17149952, 3341312, 3452160, -1698368, -438448, 534204, -23939, 133551, -27403, -26537, 13485, 3263, -4187, 231, 989, -305, -171, 119, 13, -33, 5, 7, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 3*x^34 + 7*x^33 + 5*x^32 - 33*x^31 + 13*x^30 + 119*x^29 - 171*x^28 - 305*x^27 + 989*x^26 + 231*x^25 - 4187*x^24 + 3263*x^23 + 13485*x^22 - 26537*x^21 - 27403*x^20 + 133551*x^19 - 23939*x^18 + 534204*x^17 - 438448*x^16 - 1698368*x^15 + 3452160*x^14 + 3341312*x^13 - 17149952*x^12 + 3784704*x^11 + 64815104*x^10 - 79953920*x^9 - 179306496*x^8 + 499122176*x^7 + 218103808*x^6 - 2214592512*x^5 + 1342177280*x^4 + 7516192768*x^3 - 12884901888*x^2 - 17179869184*x + 68719476736)
 
gp: K = bnfinit(x^36 - x^35 - 3*x^34 + 7*x^33 + 5*x^32 - 33*x^31 + 13*x^30 + 119*x^29 - 171*x^28 - 305*x^27 + 989*x^26 + 231*x^25 - 4187*x^24 + 3263*x^23 + 13485*x^22 - 26537*x^21 - 27403*x^20 + 133551*x^19 - 23939*x^18 + 534204*x^17 - 438448*x^16 - 1698368*x^15 + 3452160*x^14 + 3341312*x^13 - 17149952*x^12 + 3784704*x^11 + 64815104*x^10 - 79953920*x^9 - 179306496*x^8 + 499122176*x^7 + 218103808*x^6 - 2214592512*x^5 + 1342177280*x^4 + 7516192768*x^3 - 12884901888*x^2 - 17179869184*x + 68719476736, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 3 x^{34} + 7 x^{33} + 5 x^{32} - 33 x^{31} + 13 x^{30} + 119 x^{29} - 171 x^{28} - 305 x^{27} + 989 x^{26} + 231 x^{25} - 4187 x^{24} + 3263 x^{23} + 13485 x^{22} - 26537 x^{21} - 27403 x^{20} + 133551 x^{19} - 23939 x^{18} + 534204 x^{17} - 438448 x^{16} - 1698368 x^{15} + 3452160 x^{14} + 3341312 x^{13} - 17149952 x^{12} + 3784704 x^{11} + 64815104 x^{10} - 79953920 x^{9} - 179306496 x^{8} + 499122176 x^{7} + 218103808 x^{6} - 2214592512 x^{5} + 1342177280 x^{4} + 7516192768 x^{3} - 12884901888 x^{2} - 17179869184 x + 68719476736 \)

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:  \(44387950739803436916061690678602018428037077267652774810791015625=3^{18}\cdot 5^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.48$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(285=3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{285}(256,·)$, $\chi_{285}(1,·)$, $\chi_{285}(134,·)$, $\chi_{285}(136,·)$, $\chi_{285}(269,·)$, $\chi_{285}(14,·)$, $\chi_{285}(271,·)$, $\chi_{285}(16,·)$, $\chi_{285}(149,·)$, $\chi_{285}(151,·)$, $\chi_{285}(284,·)$, $\chi_{285}(29,·)$, $\chi_{285}(31,·)$, $\chi_{285}(164,·)$, $\chi_{285}(166,·)$, $\chi_{285}(44,·)$, $\chi_{285}(46,·)$, $\chi_{285}(179,·)$, $\chi_{285}(181,·)$, $\chi_{285}(59,·)$, $\chi_{285}(61,·)$, $\chi_{285}(194,·)$, $\chi_{285}(196,·)$, $\chi_{285}(74,·)$, $\chi_{285}(211,·)$, $\chi_{285}(89,·)$, $\chi_{285}(91,·)$, $\chi_{285}(224,·)$, $\chi_{285}(226,·)$, $\chi_{285}(104,·)$, $\chi_{285}(106,·)$, $\chi_{285}(239,·)$, $\chi_{285}(241,·)$, $\chi_{285}(119,·)$, $\chi_{285}(121,·)$, $\chi_{285}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{95756} a^{19} + \frac{1}{4} a^{18} - \frac{1}{4} a^{17} + \frac{1}{4} a^{16} - \frac{1}{4} a^{15} + \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{10083}{23939}$, $\frac{1}{383024} a^{20} - \frac{1}{383024} a^{19} - \frac{1}{16} a^{18} - \frac{3}{16} a^{17} + \frac{7}{16} a^{16} + \frac{5}{16} a^{15} - \frac{1}{16} a^{14} - \frac{3}{16} a^{13} + \frac{7}{16} a^{12} + \frac{5}{16} a^{11} - \frac{1}{16} a^{10} - \frac{3}{16} a^{9} + \frac{7}{16} a^{8} + \frac{5}{16} a^{7} - \frac{1}{16} a^{6} - \frac{3}{16} a^{5} + \frac{7}{16} a^{4} + \frac{5}{16} a^{3} - \frac{1}{16} a^{2} + \frac{37795}{95756} a - \frac{3464}{23939}$, $\frac{1}{1532096} a^{21} - \frac{1}{1532096} a^{20} - \frac{3}{1532096} a^{19} - \frac{19}{64} a^{18} + \frac{23}{64} a^{17} - \frac{11}{64} a^{16} - \frac{17}{64} a^{15} - \frac{3}{64} a^{14} + \frac{7}{64} a^{13} + \frac{5}{64} a^{12} + \frac{31}{64} a^{11} + \frac{13}{64} a^{10} - \frac{9}{64} a^{9} + \frac{21}{64} a^{8} + \frac{15}{64} a^{7} + \frac{29}{64} a^{6} - \frac{25}{64} a^{5} - \frac{27}{64} a^{4} - \frac{1}{64} a^{3} + \frac{133551}{383024} a^{2} - \frac{27403}{95756} a - \frac{2598}{23939}$, $\frac{1}{6128384} a^{22} - \frac{1}{6128384} a^{21} - \frac{3}{6128384} a^{20} + \frac{7}{6128384} a^{19} - \frac{41}{256} a^{18} + \frac{117}{256} a^{17} + \frac{47}{256} a^{16} - \frac{3}{256} a^{15} + \frac{71}{256} a^{14} - \frac{59}{256} a^{13} + \frac{31}{256} a^{12} - \frac{51}{256} a^{11} - \frac{73}{256} a^{10} + \frac{21}{256} a^{9} + \frac{15}{256} a^{8} - \frac{99}{256} a^{7} + \frac{39}{256} a^{6} + \frac{101}{256} a^{5} - \frac{1}{256} a^{4} + \frac{133551}{1532096} a^{3} - \frac{27403}{383024} a^{2} - \frac{26537}{95756} a - \frac{10454}{23939}$, $\frac{1}{24513536} a^{23} - \frac{1}{24513536} a^{22} - \frac{3}{24513536} a^{21} + \frac{7}{24513536} a^{20} + \frac{5}{24513536} a^{19} + \frac{373}{1024} a^{18} - \frac{209}{1024} a^{17} - \frac{259}{1024} a^{16} + \frac{71}{1024} a^{15} - \frac{59}{1024} a^{14} - \frac{225}{1024} a^{13} + \frac{461}{1024} a^{12} + \frac{439}{1024} a^{11} - \frac{235}{1024} a^{10} - \frac{497}{1024} a^{9} + \frac{413}{1024} a^{8} - \frac{473}{1024} a^{7} - \frac{155}{1024} a^{6} - \frac{1}{1024} a^{5} + \frac{133551}{6128384} a^{4} - \frac{27403}{1532096} a^{3} - \frac{26537}{383024} a^{2} + \frac{13485}{95756} a + \frac{3263}{23939}$, $\frac{1}{98054144} a^{24} - \frac{1}{98054144} a^{23} - \frac{3}{98054144} a^{22} + \frac{7}{98054144} a^{21} + \frac{5}{98054144} a^{20} - \frac{33}{98054144} a^{19} + \frac{1839}{4096} a^{18} + \frac{765}{4096} a^{17} + \frac{71}{4096} a^{16} + \frac{965}{4096} a^{15} - \frac{1249}{4096} a^{14} + \frac{1485}{4096} a^{13} - \frac{585}{4096} a^{12} - \frac{1259}{4096} a^{11} - \frac{497}{4096} a^{10} + \frac{1437}{4096} a^{9} + \frac{551}{4096} a^{8} + \frac{1893}{4096} a^{7} - \frac{1}{4096} a^{6} + \frac{133551}{24513536} a^{5} - \frac{27403}{6128384} a^{4} - \frac{26537}{1532096} a^{3} + \frac{13485}{383024} a^{2} + \frac{3263}{95756} a - \frac{4187}{23939}$, $\frac{1}{392216576} a^{25} - \frac{1}{392216576} a^{24} - \frac{3}{392216576} a^{23} + \frac{7}{392216576} a^{22} + \frac{5}{392216576} a^{21} - \frac{33}{392216576} a^{20} + \frac{13}{392216576} a^{19} - \frac{3331}{16384} a^{18} - \frac{4025}{16384} a^{17} + \frac{965}{16384} a^{16} - \frac{1249}{16384} a^{15} - \frac{2611}{16384} a^{14} + \frac{7607}{16384} a^{13} + \frac{2837}{16384} a^{12} - \frac{497}{16384} a^{11} + \frac{5533}{16384} a^{10} - \frac{3545}{16384} a^{9} - \frac{2203}{16384} a^{8} - \frac{1}{16384} a^{7} + \frac{133551}{98054144} a^{6} - \frac{27403}{24513536} a^{5} - \frac{26537}{6128384} a^{4} + \frac{13485}{1532096} a^{3} + \frac{3263}{383024} a^{2} - \frac{4187}{95756} a + \frac{231}{23939}$, $\frac{1}{1568866304} a^{26} - \frac{1}{1568866304} a^{25} - \frac{3}{1568866304} a^{24} + \frac{7}{1568866304} a^{23} + \frac{5}{1568866304} a^{22} - \frac{33}{1568866304} a^{21} + \frac{13}{1568866304} a^{20} + \frac{119}{1568866304} a^{19} - \frac{20409}{65536} a^{18} - \frac{31803}{65536} a^{17} - \frac{17633}{65536} a^{16} + \frac{13773}{65536} a^{15} - \frac{8777}{65536} a^{14} + \frac{19221}{65536} a^{13} + \frac{15887}{65536} a^{12} - \frac{27235}{65536} a^{11} + \frac{29223}{65536} a^{10} + \frac{14181}{65536} a^{9} - \frac{1}{65536} a^{8} + \frac{133551}{392216576} a^{7} - \frac{27403}{98054144} a^{6} - \frac{26537}{24513536} a^{5} + \frac{13485}{6128384} a^{4} + \frac{3263}{1532096} a^{3} - \frac{4187}{383024} a^{2} + \frac{231}{95756} a + \frac{989}{23939}$, $\frac{1}{6275465216} a^{27} - \frac{1}{6275465216} a^{26} - \frac{3}{6275465216} a^{25} + \frac{7}{6275465216} a^{24} + \frac{5}{6275465216} a^{23} - \frac{33}{6275465216} a^{22} + \frac{13}{6275465216} a^{21} + \frac{119}{6275465216} a^{20} - \frac{171}{6275465216} a^{19} - \frac{97339}{262144} a^{18} - \frac{83169}{262144} a^{17} - \frac{51763}{262144} a^{16} + \frac{122295}{262144} a^{15} + \frac{84757}{262144} a^{14} - \frac{49649}{262144} a^{13} - \frac{27235}{262144} a^{12} - \frac{36313}{262144} a^{11} - \frac{116891}{262144} a^{10} - \frac{1}{262144} a^{9} + \frac{133551}{1568866304} a^{8} - \frac{27403}{392216576} a^{7} - \frac{26537}{98054144} a^{6} + \frac{13485}{24513536} a^{5} + \frac{3263}{6128384} a^{4} - \frac{4187}{1532096} a^{3} + \frac{231}{383024} a^{2} + \frac{989}{95756} a - \frac{305}{23939}$, $\frac{1}{25101860864} a^{28} - \frac{1}{25101860864} a^{27} - \frac{3}{25101860864} a^{26} + \frac{7}{25101860864} a^{25} + \frac{5}{25101860864} a^{24} - \frac{33}{25101860864} a^{23} + \frac{13}{25101860864} a^{22} + \frac{119}{25101860864} a^{21} - \frac{171}{25101860864} a^{20} - \frac{305}{25101860864} a^{19} + \frac{178975}{1048576} a^{18} + \frac{210381}{1048576} a^{17} + \frac{122295}{1048576} a^{16} + \frac{84757}{1048576} a^{15} + \frac{474639}{1048576} a^{14} + \frac{234909}{1048576} a^{13} - \frac{36313}{1048576} a^{12} + \frac{145253}{1048576} a^{11} - \frac{1}{1048576} a^{10} + \frac{133551}{6275465216} a^{9} - \frac{27403}{1568866304} a^{8} - \frac{26537}{392216576} a^{7} + \frac{13485}{98054144} a^{6} + \frac{3263}{24513536} a^{5} - \frac{4187}{6128384} a^{4} + \frac{231}{1532096} a^{3} + \frac{989}{383024} a^{2} - \frac{305}{95756} a - \frac{171}{23939}$, $\frac{1}{100407443456} a^{29} - \frac{1}{100407443456} a^{28} - \frac{3}{100407443456} a^{27} + \frac{7}{100407443456} a^{26} + \frac{5}{100407443456} a^{25} - \frac{33}{100407443456} a^{24} + \frac{13}{100407443456} a^{23} + \frac{119}{100407443456} a^{22} - \frac{171}{100407443456} a^{21} - \frac{305}{100407443456} a^{20} + \frac{989}{100407443456} a^{19} - \frac{838195}{4194304} a^{18} + \frac{122295}{4194304} a^{17} - \frac{963819}{4194304} a^{16} + \frac{474639}{4194304} a^{15} - \frac{813667}{4194304} a^{14} - \frac{1084889}{4194304} a^{13} + \frac{145253}{4194304} a^{12} - \frac{1}{4194304} a^{11} + \frac{133551}{25101860864} a^{10} - \frac{27403}{6275465216} a^{9} - \frac{26537}{1568866304} a^{8} + \frac{13485}{392216576} a^{7} + \frac{3263}{98054144} a^{6} - \frac{4187}{24513536} a^{5} + \frac{231}{6128384} a^{4} + \frac{989}{1532096} a^{3} - \frac{305}{383024} a^{2} - \frac{171}{95756} a + \frac{119}{23939}$, $\frac{1}{401629773824} a^{30} - \frac{1}{401629773824} a^{29} - \frac{3}{401629773824} a^{28} + \frac{7}{401629773824} a^{27} + \frac{5}{401629773824} a^{26} - \frac{33}{401629773824} a^{25} + \frac{13}{401629773824} a^{24} + \frac{119}{401629773824} a^{23} - \frac{171}{401629773824} a^{22} - \frac{305}{401629773824} a^{21} + \frac{989}{401629773824} a^{20} + \frac{231}{401629773824} a^{19} - \frac{8266313}{16777216} a^{18} - \frac{5158123}{16777216} a^{17} + \frac{4668943}{16777216} a^{16} - \frac{813667}{16777216} a^{15} - \frac{1084889}{16777216} a^{14} + \frac{4339557}{16777216} a^{13} - \frac{1}{16777216} a^{12} + \frac{133551}{100407443456} a^{11} - \frac{27403}{25101860864} a^{10} - \frac{26537}{6275465216} a^{9} + \frac{13485}{1568866304} a^{8} + \frac{3263}{392216576} a^{7} - \frac{4187}{98054144} a^{6} + \frac{231}{24513536} a^{5} + \frac{989}{6128384} a^{4} - \frac{305}{1532096} a^{3} - \frac{171}{383024} a^{2} + \frac{119}{95756} a + \frac{13}{23939}$, $\frac{1}{1606519095296} a^{31} - \frac{1}{1606519095296} a^{30} - \frac{3}{1606519095296} a^{29} + \frac{7}{1606519095296} a^{28} + \frac{5}{1606519095296} a^{27} - \frac{33}{1606519095296} a^{26} + \frac{13}{1606519095296} a^{25} + \frac{119}{1606519095296} a^{24} - \frac{171}{1606519095296} a^{23} - \frac{305}{1606519095296} a^{22} + \frac{989}{1606519095296} a^{21} + \frac{231}{1606519095296} a^{20} - \frac{4187}{1606519095296} a^{19} - \frac{5158123}{67108864} a^{18} - \frac{28885489}{67108864} a^{17} - \frac{17590883}{67108864} a^{16} - \frac{1084889}{67108864} a^{15} + \frac{4339557}{67108864} a^{14} - \frac{1}{67108864} a^{13} + \frac{133551}{401629773824} a^{12} - \frac{27403}{100407443456} a^{11} - \frac{26537}{25101860864} a^{10} + \frac{13485}{6275465216} a^{9} + \frac{3263}{1568866304} a^{8} - \frac{4187}{392216576} a^{7} + \frac{231}{98054144} a^{6} + \frac{989}{24513536} a^{5} - \frac{305}{6128384} a^{4} - \frac{171}{1532096} a^{3} + \frac{119}{383024} a^{2} + \frac{13}{95756} a - \frac{33}{23939}$, $\frac{1}{6426076381184} a^{32} - \frac{1}{6426076381184} a^{31} - \frac{3}{6426076381184} a^{30} + \frac{7}{6426076381184} a^{29} + \frac{5}{6426076381184} a^{28} - \frac{33}{6426076381184} a^{27} + \frac{13}{6426076381184} a^{26} + \frac{119}{6426076381184} a^{25} - \frac{171}{6426076381184} a^{24} - \frac{305}{6426076381184} a^{23} + \frac{989}{6426076381184} a^{22} + \frac{231}{6426076381184} a^{21} - \frac{4187}{6426076381184} a^{20} + \frac{3263}{6426076381184} a^{19} - \frac{28885489}{268435456} a^{18} + \frac{49517981}{268435456} a^{17} + \frac{66023975}{268435456} a^{16} + \frac{4339557}{268435456} a^{15} - \frac{1}{268435456} a^{14} + \frac{133551}{1606519095296} a^{13} - \frac{27403}{401629773824} a^{12} - \frac{26537}{100407443456} a^{11} + \frac{13485}{25101860864} a^{10} + \frac{3263}{6275465216} a^{9} - \frac{4187}{1568866304} a^{8} + \frac{231}{392216576} a^{7} + \frac{989}{98054144} a^{6} - \frac{305}{24513536} a^{5} - \frac{171}{6128384} a^{4} + \frac{119}{1532096} a^{3} + \frac{13}{383024} a^{2} - \frac{33}{95756} a + \frac{5}{23939}$, $\frac{1}{25704305524736} a^{33} - \frac{1}{25704305524736} a^{32} - \frac{3}{25704305524736} a^{31} + \frac{7}{25704305524736} a^{30} + \frac{5}{25704305524736} a^{29} - \frac{33}{25704305524736} a^{28} + \frac{13}{25704305524736} a^{27} + \frac{119}{25704305524736} a^{26} - \frac{171}{25704305524736} a^{25} - \frac{305}{25704305524736} a^{24} + \frac{989}{25704305524736} a^{23} + \frac{231}{25704305524736} a^{22} - \frac{4187}{25704305524736} a^{21} + \frac{3263}{25704305524736} a^{20} + \frac{13485}{25704305524736} a^{19} + \frac{49517981}{1073741824} a^{18} + \frac{66023975}{1073741824} a^{17} - \frac{264095899}{1073741824} a^{16} - \frac{1}{1073741824} a^{15} + \frac{133551}{6426076381184} a^{14} - \frac{27403}{1606519095296} a^{13} - \frac{26537}{401629773824} a^{12} + \frac{13485}{100407443456} a^{11} + \frac{3263}{25101860864} a^{10} - \frac{4187}{6275465216} a^{9} + \frac{231}{1568866304} a^{8} + \frac{989}{392216576} a^{7} - \frac{305}{98054144} a^{6} - \frac{171}{24513536} a^{5} + \frac{119}{6128384} a^{4} + \frac{13}{1532096} a^{3} - \frac{33}{383024} a^{2} + \frac{5}{95756} a + \frac{7}{23939}$, $\frac{1}{102817222098944} a^{34} - \frac{1}{102817222098944} a^{33} - \frac{3}{102817222098944} a^{32} + \frac{7}{102817222098944} a^{31} + \frac{5}{102817222098944} a^{30} - \frac{33}{102817222098944} a^{29} + \frac{13}{102817222098944} a^{28} + \frac{119}{102817222098944} a^{27} - \frac{171}{102817222098944} a^{26} - \frac{305}{102817222098944} a^{25} + \frac{989}{102817222098944} a^{24} + \frac{231}{102817222098944} a^{23} - \frac{4187}{102817222098944} a^{22} + \frac{3263}{102817222098944} a^{21} + \frac{13485}{102817222098944} a^{20} - \frac{26537}{102817222098944} a^{19} + \frac{66023975}{4294967296} a^{18} - \frac{264095899}{4294967296} a^{17} - \frac{1}{4294967296} a^{16} + \frac{133551}{25704305524736} a^{15} - \frac{27403}{6426076381184} a^{14} - \frac{26537}{1606519095296} a^{13} + \frac{13485}{401629773824} a^{12} + \frac{3263}{100407443456} a^{11} - \frac{4187}{25101860864} a^{10} + \frac{231}{6275465216} a^{9} + \frac{989}{1568866304} a^{8} - \frac{305}{392216576} a^{7} - \frac{171}{98054144} a^{6} + \frac{119}{24513536} a^{5} + \frac{13}{6128384} a^{4} - \frac{33}{1532096} a^{3} + \frac{5}{383024} a^{2} + \frac{7}{95756} a - \frac{3}{23939}$, $\frac{1}{411268888395776} a^{35} - \frac{1}{411268888395776} a^{34} - \frac{3}{411268888395776} a^{33} + \frac{7}{411268888395776} a^{32} + \frac{5}{411268888395776} a^{31} - \frac{33}{411268888395776} a^{30} + \frac{13}{411268888395776} a^{29} + \frac{119}{411268888395776} a^{28} - \frac{171}{411268888395776} a^{27} - \frac{305}{411268888395776} a^{26} + \frac{989}{411268888395776} a^{25} + \frac{231}{411268888395776} a^{24} - \frac{4187}{411268888395776} a^{23} + \frac{3263}{411268888395776} a^{22} + \frac{13485}{411268888395776} a^{21} - \frac{26537}{411268888395776} a^{20} - \frac{27403}{411268888395776} a^{19} - \frac{264095899}{17179869184} a^{18} - \frac{1}{17179869184} a^{17} + \frac{133551}{102817222098944} a^{16} - \frac{27403}{25704305524736} a^{15} - \frac{26537}{6426076381184} a^{14} + \frac{13485}{1606519095296} a^{13} + \frac{3263}{401629773824} a^{12} - \frac{4187}{100407443456} a^{11} + \frac{231}{25101860864} a^{10} + \frac{989}{6275465216} a^{9} - \frac{305}{1568866304} a^{8} - \frac{171}{392216576} a^{7} + \frac{119}{98054144} a^{6} + \frac{13}{24513536} a^{5} - \frac{33}{6128384} a^{4} + \frac{5}{1532096} a^{3} + \frac{7}{383024} a^{2} - \frac{3}{95756} a - \frac{1}{23939}$

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:  \( -\frac{231}{401629773824} a^{31} - \frac{23168531}{401629773824} a^{12} \) (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

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{285}) \), 3.3.361.1, \(\Q(\sqrt{-15}, \sqrt{-19})\), 6.0.2476099.1, 6.0.439833375.1, 6.6.8356834125.1, \(\Q(\zeta_{19})^+\), 12.0.69836676592764515625.1, \(\Q(\zeta_{19})\), 18.0.11088656920413061413017818359375.2, 18.18.210684481487848166847338548828125.1

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 $18^{2}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{4}$ 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
3Data not computed
5Data not computed
19Data not computed