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Properties

Label 36.0.52488303469...2921.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 7^{18}\cdot 19^{32}$
Root discriminant $62.77$
Ramified primes $3, 7, 19$
Class number $1332$ (GRH)
Class group $[2, 666]$ (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, -655360, -3604480, -13107200, 88227840, 80846848, -296185856, 37050368, 631394304, -223400960, -784816896, 105430656, 716973312, -25242848, -460870720, 36478832, 219610804, -21277922, -82683121, 5036015, 25482307, -1034028, -6307141, 260669, 1281217, -33939, -221058, 708, 31599, 75, -3596, -35, 338, 12, -25, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 25*x^34 + 12*x^33 + 338*x^32 - 35*x^31 - 3596*x^30 + 75*x^29 + 31599*x^28 + 708*x^27 - 221058*x^26 - 33939*x^25 + 1281217*x^24 + 260669*x^23 - 6307141*x^22 - 1034028*x^21 + 25482307*x^20 + 5036015*x^19 - 82683121*x^18 - 21277922*x^17 + 219610804*x^16 + 36478832*x^15 - 460870720*x^14 - 25242848*x^13 + 716973312*x^12 + 105430656*x^11 - 784816896*x^10 - 223400960*x^9 + 631394304*x^8 + 37050368*x^7 - 296185856*x^6 + 80846848*x^5 + 88227840*x^4 - 13107200*x^3 - 3604480*x^2 - 655360*x + 262144)
 
gp: K = bnfinit(x^36 - x^35 - 25*x^34 + 12*x^33 + 338*x^32 - 35*x^31 - 3596*x^30 + 75*x^29 + 31599*x^28 + 708*x^27 - 221058*x^26 - 33939*x^25 + 1281217*x^24 + 260669*x^23 - 6307141*x^22 - 1034028*x^21 + 25482307*x^20 + 5036015*x^19 - 82683121*x^18 - 21277922*x^17 + 219610804*x^16 + 36478832*x^15 - 460870720*x^14 - 25242848*x^13 + 716973312*x^12 + 105430656*x^11 - 784816896*x^10 - 223400960*x^9 + 631394304*x^8 + 37050368*x^7 - 296185856*x^6 + 80846848*x^5 + 88227840*x^4 - 13107200*x^3 - 3604480*x^2 - 655360*x + 262144, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 25 x^{34} + 12 x^{33} + 338 x^{32} - 35 x^{31} - 3596 x^{30} + 75 x^{29} + 31599 x^{28} + 708 x^{27} - 221058 x^{26} - 33939 x^{25} + 1281217 x^{24} + 260669 x^{23} - 6307141 x^{22} - 1034028 x^{21} + 25482307 x^{20} + 5036015 x^{19} - 82683121 x^{18} - 21277922 x^{17} + 219610804 x^{16} + 36478832 x^{15} - 460870720 x^{14} - 25242848 x^{13} + 716973312 x^{12} + 105430656 x^{11} - 784816896 x^{10} - 223400960 x^{9} + 631394304 x^{8} + 37050368 x^{7} - 296185856 x^{6} + 80846848 x^{5} + 88227840 x^{4} - 13107200 x^{3} - 3604480 x^{2} - 655360 x + 262144 \)

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:  \(52488303469710461753453989107747724546083978878098845653195292921=3^{18}\cdot 7^{18}\cdot 19^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.77$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(399=3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{399}(1,·)$, $\chi_{399}(386,·)$, $\chi_{399}(134,·)$, $\chi_{399}(391,·)$, $\chi_{399}(139,·)$, $\chi_{399}(272,·)$, $\chi_{399}(20,·)$, $\chi_{399}(286,·)$, $\chi_{399}(169,·)$, $\chi_{399}(43,·)$, $\chi_{399}(302,·)$, $\chi_{399}(125,·)$, $\chi_{399}(176,·)$, $\chi_{399}(55,·)$, $\chi_{399}(188,·)$, $\chi_{399}(62,·)$, $\chi_{399}(64,·)$, $\chi_{399}(197,·)$, $\chi_{399}(328,·)$, $\chi_{399}(83,·)$, $\chi_{399}(85,·)$, $\chi_{399}(218,·)$, $\chi_{399}(92,·)$, $\chi_{399}(349,·)$, $\chi_{399}(358,·)$, $\chi_{399}(104,·)$, $\chi_{399}(106,·)$, $\chi_{399}(365,·)$, $\chi_{399}(239,·)$, $\chi_{399}(232,·)$, $\chi_{399}(370,·)$, $\chi_{399}(244,·)$, $\chi_{399}(118,·)$, $\chi_{399}(377,·)$, $\chi_{399}(251,·)$, $\chi_{399}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{19} - \frac{1}{4} a^{18} - \frac{1}{2} a^{16} + \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{21} - \frac{1}{8} a^{20} - \frac{1}{8} a^{19} - \frac{1}{2} a^{18} + \frac{1}{4} a^{17} - \frac{3}{8} a^{16} - \frac{1}{2} a^{15} + \frac{3}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} - \frac{3}{8} a^{10} + \frac{1}{8} a^{9} - \frac{3}{8} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{22} - \frac{1}{16} a^{21} - \frac{1}{16} a^{20} - \frac{1}{4} a^{19} + \frac{1}{8} a^{18} + \frac{5}{16} a^{17} + \frac{1}{4} a^{16} + \frac{3}{16} a^{15} + \frac{7}{16} a^{14} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} + \frac{5}{16} a^{11} + \frac{1}{16} a^{10} + \frac{5}{16} a^{9} - \frac{5}{16} a^{8} + \frac{1}{4} a^{7} + \frac{3}{16} a^{6} + \frac{7}{16} a^{5} + \frac{7}{16} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{23} - \frac{1}{32} a^{22} - \frac{1}{32} a^{21} - \frac{1}{8} a^{20} + \frac{1}{16} a^{19} + \frac{5}{32} a^{18} - \frac{3}{8} a^{17} - \frac{13}{32} a^{16} - \frac{9}{32} a^{15} - \frac{1}{8} a^{14} - \frac{1}{16} a^{13} - \frac{11}{32} a^{12} + \frac{1}{32} a^{11} + \frac{5}{32} a^{10} - \frac{5}{32} a^{9} + \frac{1}{8} a^{8} + \frac{3}{32} a^{7} - \frac{9}{32} a^{6} + \frac{7}{32} a^{5} - \frac{1}{16} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{64} a^{24} - \frac{1}{64} a^{23} - \frac{1}{64} a^{22} - \frac{1}{16} a^{21} + \frac{1}{32} a^{20} + \frac{5}{64} a^{19} + \frac{5}{16} a^{18} - \frac{13}{64} a^{17} + \frac{23}{64} a^{16} - \frac{1}{16} a^{15} + \frac{15}{32} a^{14} - \frac{11}{64} a^{13} + \frac{1}{64} a^{12} + \frac{5}{64} a^{11} + \frac{27}{64} a^{10} - \frac{7}{16} a^{9} - \frac{29}{64} a^{8} + \frac{23}{64} a^{7} + \frac{7}{64} a^{6} - \frac{1}{32} a^{5} + \frac{1}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{128} a^{25} - \frac{1}{128} a^{24} - \frac{1}{128} a^{23} - \frac{1}{32} a^{22} + \frac{1}{64} a^{21} + \frac{5}{128} a^{20} + \frac{5}{32} a^{19} - \frac{13}{128} a^{18} - \frac{41}{128} a^{17} - \frac{1}{32} a^{16} - \frac{17}{64} a^{15} - \frac{11}{128} a^{14} - \frac{63}{128} a^{13} - \frac{59}{128} a^{12} + \frac{27}{128} a^{11} + \frac{9}{32} a^{10} + \frac{35}{128} a^{9} - \frac{41}{128} a^{8} + \frac{7}{128} a^{7} - \frac{1}{64} a^{6} - \frac{15}{32} a^{5} - \frac{7}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{9472} a^{26} + \frac{13}{9472} a^{25} - \frac{67}{9472} a^{24} - \frac{35}{4736} a^{23} - \frac{141}{4736} a^{22} + \frac{5}{256} a^{21} + \frac{281}{4736} a^{20} - \frac{553}{9472} a^{19} - \frac{2167}{9472} a^{18} - \frac{383}{4736} a^{17} - \frac{559}{4736} a^{16} + \frac{1937}{9472} a^{15} + \frac{1487}{9472} a^{14} - \frac{2305}{9472} a^{13} + \frac{3621}{9472} a^{12} + \frac{1529}{4736} a^{11} - \frac{1417}{9472} a^{10} + \frac{2009}{9472} a^{9} + \frac{237}{9472} a^{8} - \frac{537}{2368} a^{7} + \frac{473}{2368} a^{6} + \frac{7}{16} a^{5} - \frac{181}{592} a^{4} + \frac{23}{296} a^{3} - \frac{14}{37} a^{2} + \frac{8}{37} a + \frac{2}{37}$, $\frac{1}{18944} a^{27} - \frac{1}{18944} a^{26} + \frac{47}{18944} a^{25} - \frac{5}{4736} a^{24} - \frac{95}{9472} a^{23} - \frac{11}{18944} a^{22} - \frac{63}{4736} a^{21} - \frac{1021}{18944} a^{20} + \frac{3799}{18944} a^{19} + \frac{1991}{4736} a^{18} - \frac{4521}{9472} a^{17} - \frac{7867}{18944} a^{16} + \frac{1009}{18944} a^{15} + \frac{853}{18944} a^{14} - \frac{4661}{18944} a^{13} + \frac{1041}{4736} a^{12} - \frac{2493}{18944} a^{11} - \frac{1241}{18944} a^{10} - \frac{6873}{18944} a^{9} + \frac{967}{9472} a^{8} + \frac{1257}{4736} a^{7} - \frac{277}{2368} a^{6} - \frac{477}{1184} a^{5} + \frac{127}{296} a^{4} + \frac{21}{148} a^{3} - \frac{73}{148} a^{2} - \frac{18}{37} a - \frac{14}{37}$, $\frac{1}{37888} a^{28} - \frac{1}{37888} a^{27} - \frac{1}{37888} a^{26} - \frac{13}{9472} a^{25} + \frac{33}{18944} a^{24} + \frac{389}{37888} a^{23} - \frac{231}{9472} a^{22} + \frac{755}{37888} a^{21} - \frac{1273}{37888} a^{20} - \frac{2029}{9472} a^{19} - \frac{1353}{18944} a^{18} - \frac{11947}{37888} a^{17} + \frac{16785}{37888} a^{16} + \frac{15621}{37888} a^{15} + \frac{2699}{37888} a^{14} - \frac{3711}{9472} a^{13} + \frac{8995}{37888} a^{12} + \frac{12407}{37888} a^{11} - \frac{9897}{37888} a^{10} + \frac{155}{512} a^{9} + \frac{3593}{9472} a^{8} + \frac{401}{4736} a^{7} - \frac{1047}{2368} a^{6} + \frac{119}{296} a^{5} - \frac{3}{296} a^{4} + \frac{33}{74} a^{3} - \frac{67}{148} a^{2} + \frac{8}{37} a + \frac{13}{37}$, $\frac{1}{75776} a^{29} - \frac{1}{75776} a^{28} - \frac{1}{75776} a^{27} - \frac{1}{18944} a^{26} + \frac{49}{37888} a^{25} + \frac{133}{75776} a^{24} + \frac{261}{18944} a^{23} - \frac{941}{75776} a^{22} - \frac{681}{75776} a^{21} + \frac{423}{18944} a^{20} + \frac{6687}{37888} a^{19} - \frac{6443}{75776} a^{18} + \frac{1921}{75776} a^{17} + \frac{30629}{75776} a^{16} + \frac{4507}{75776} a^{15} + \frac{5105}{18944} a^{14} + \frac{37475}{75776} a^{13} + \frac{12759}{75776} a^{12} - \frac{11705}{75776} a^{11} - \frac{3409}{37888} a^{10} - \frac{5303}{18944} a^{9} + \frac{3377}{9472} a^{8} + \frac{565}{2368} a^{7} - \frac{497}{2368} a^{6} + \frac{17}{296} a^{5} - \frac{33}{296} a^{4} + \frac{145}{296} a^{3} + \frac{13}{148} a^{2} - \frac{1}{37} a + \frac{12}{37}$, $\frac{1}{151552} a^{30} - \frac{1}{151552} a^{29} - \frac{1}{151552} a^{28} - \frac{1}{37888} a^{27} + \frac{1}{75776} a^{26} + \frac{69}{151552} a^{25} - \frac{203}{37888} a^{24} + \frac{2227}{151552} a^{23} + \frac{343}{151552} a^{22} + \frac{1311}{37888} a^{21} + \frac{3983}{75776} a^{20} - \frac{3083}{151552} a^{19} - \frac{70655}{151552} a^{18} - \frac{41467}{151552} a^{17} + \frac{10011}{151552} a^{16} - \frac{13559}{37888} a^{15} - \frac{47261}{151552} a^{14} + \frac{10263}{151552} a^{13} + \frac{46791}{151552} a^{12} + \frac{32735}{75776} a^{11} + \frac{12721}{37888} a^{10} - \frac{6079}{18944} a^{9} - \frac{1737}{4736} a^{8} + \frac{497}{1184} a^{7} - \frac{119}{592} a^{6} + \frac{39}{296} a^{5} + \frac{121}{592} a^{4} - \frac{125}{296} a^{3} + \frac{1}{148} a^{2} + \frac{27}{74} a - \frac{12}{37}$, $\frac{1}{303104} a^{31} - \frac{1}{303104} a^{30} - \frac{1}{303104} a^{29} - \frac{1}{75776} a^{28} + \frac{1}{151552} a^{27} + \frac{5}{303104} a^{26} + \frac{181}{75776} a^{25} - \frac{589}{303104} a^{24} - \frac{2281}{303104} a^{23} + \frac{2271}{75776} a^{22} + \frac{5167}{151552} a^{21} - \frac{17739}{303104} a^{20} + \frac{7361}{303104} a^{19} - \frac{37755}{303104} a^{18} - \frac{33317}{303104} a^{17} - \frac{619}{2048} a^{16} - \frac{128605}{303104} a^{15} + \frac{144791}{303104} a^{14} + \frac{2503}{303104} a^{13} - \frac{46433}{151552} a^{12} - \frac{33247}{75776} a^{11} + \frac{5849}{37888} a^{10} - \frac{4053}{9472} a^{9} + \frac{249}{2368} a^{8} + \frac{377}{4736} a^{7} + \frac{57}{148} a^{6} + \frac{153}{592} a^{5} + \frac{237}{592} a^{4} + \frac{103}{296} a^{3} - \frac{9}{148} a^{2} - \frac{7}{74} a - \frac{4}{37}$, $\frac{1}{606208} a^{32} - \frac{1}{606208} a^{31} - \frac{1}{606208} a^{30} - \frac{1}{151552} a^{29} + \frac{1}{303104} a^{28} + \frac{5}{606208} a^{27} + \frac{5}{151552} a^{26} - \frac{269}{606208} a^{25} - \frac{2473}{606208} a^{24} - \frac{1985}{151552} a^{23} + \frac{239}{303104} a^{22} - \frac{34315}{606208} a^{21} - \frac{37823}{606208} a^{20} - \frac{140987}{606208} a^{19} - \frac{13797}{606208} a^{18} + \frac{43241}{151552} a^{17} + \frac{222755}{606208} a^{16} - \frac{82217}{606208} a^{15} - \frac{239225}{606208} a^{14} - \frac{139649}{303104} a^{13} + \frac{9073}{151552} a^{12} - \frac{43}{2048} a^{11} - \frac{115}{512} a^{10} + \frac{2819}{9472} a^{9} - \frac{893}{4736} a^{8} + \frac{231}{592} a^{7} + \frac{283}{2368} a^{6} - \frac{159}{592} a^{5} - \frac{163}{592} a^{4} - \frac{3}{296} a^{3} + \frac{17}{148} a^{2} + \frac{5}{74} a + \frac{15}{37}$, $\frac{1}{1212416} a^{33} - \frac{1}{1212416} a^{32} - \frac{1}{1212416} a^{31} - \frac{1}{303104} a^{30} + \frac{1}{606208} a^{29} + \frac{5}{1212416} a^{28} + \frac{5}{303104} a^{27} - \frac{13}{1212416} a^{26} + \frac{855}{1212416} a^{25} - \frac{1537}{303104} a^{24} + \frac{751}{606208} a^{23} - \frac{11787}{1212416} a^{22} + \frac{47425}{1212416} a^{21} + \frac{40773}{1212416} a^{20} - \frac{287973}{1212416} a^{19} + \frac{122409}{303104} a^{18} - \frac{522717}{1212416} a^{17} + \frac{484055}{1212416} a^{16} - \frac{311673}{1212416} a^{15} - \frac{271361}{606208} a^{14} + \frac{17841}{303104} a^{13} - \frac{49111}{151552} a^{12} - \frac{8799}{37888} a^{11} - \frac{101}{512} a^{10} - \frac{3683}{9472} a^{9} + \frac{199}{4736} a^{8} - \frac{903}{4736} a^{7} + \frac{1109}{2368} a^{6} + \frac{59}{1184} a^{5} + \frac{5}{592} a^{4} - \frac{11}{296} a^{3} + \frac{1}{37} a^{2} + \frac{5}{37} a + \frac{4}{37}$, $\frac{1}{130218701754670953581314288970841513721856} a^{34} - \frac{48349456492983309970163336057537599}{130218701754670953581314288970841513721856} a^{33} - \frac{105375220060045303346863553524306627}{130218701754670953581314288970841513721856} a^{32} + \frac{70209623116744854085408643128225413}{65109350877335476790657144485420756860928} a^{31} + \frac{179541429041531743287178646325992437}{65109350877335476790657144485420756860928} a^{30} - \frac{353035699862342490819220947877731751}{130218701754670953581314288970841513721856} a^{29} + \frac{594521000206231791581859564737291423}{65109350877335476790657144485420756860928} a^{28} - \frac{498984710734326335895852357198185829}{130218701754670953581314288970841513721856} a^{27} - \frac{4608410791512274131490720879952387123}{130218701754670953581314288970841513721856} a^{26} + \frac{166418428904050102729366660397596784725}{65109350877335476790657144485420756860928} a^{25} - \frac{314784252692288576373868737523481474221}{65109350877335476790657144485420756860928} a^{24} - \frac{1351643983971934203689489220872477789343}{130218701754670953581314288970841513721856} a^{23} - \frac{3729417029363629608534332674998031298805}{130218701754670953581314288970841513721856} a^{22} + \frac{6761214255789018470963781994422573881799}{130218701754670953581314288970841513721856} a^{21} + \frac{9283931508271751562390752130404037244597}{130218701754670953581314288970841513721856} a^{20} + \frac{14651275807452227901361343948966529300213}{65109350877335476790657144485420756860928} a^{19} + \frac{28109493719344096804546547898536251804827}{130218701754670953581314288970841513721856} a^{18} + \frac{47283933135899883942843880094244436353325}{130218701754670953581314288970841513721856} a^{17} + \frac{56852179530615722886331218365180563984789}{130218701754670953581314288970841513721856} a^{16} + \frac{939878307075204975454298673019249111399}{32554675438667738395328572242710378430464} a^{15} - \frac{3525982335637835111679744466845197362273}{8138668859666934598832143060677594607616} a^{14} - \frac{1361979783688380237001051672684867722169}{4069334429833467299416071530338797303808} a^{13} + \frac{158616427524207706343650320902237493}{476977604153251749330841180371423232} a^{12} - \frac{1347165482152861716602573416988297390139}{4069334429833467299416071530338797303808} a^{11} + \frac{202507347509407519579958484216196878607}{508666803729183412427008941292349662976} a^{10} + \frac{48850901347425383275067458026724140167}{127166700932295853106752235323087415744} a^{9} - \frac{245276799735282590261691487721027798361}{508666803729183412427008941292349662976} a^{8} + \frac{93731024969611496973867342986617909471}{254333401864591706213504470646174831488} a^{7} - \frac{1161884239711015219980995060604508097}{3436937863035023056939249603326686912} a^{6} + \frac{2909642210563288709879860648153639513}{31791675233073963276688058830771853936} a^{5} - \frac{8043045431003976780140575726099674107}{31791675233073963276688058830771853936} a^{4} + \frac{73065114817855301339330334906289560}{1986979702067122704793003676923240871} a^{3} + \frac{379002777059577218420578874815645561}{7947918808268490819172014707692963484} a^{2} + \frac{362630274105544872397304121111018497}{1986979702067122704793003676923240871} a + \frac{907523104128544495467277648124293466}{1986979702067122704793003676923240871}$, $\frac{1}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{35} - \frac{5396585976338383621698463888938540799927775291}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{34} - \frac{355809816386694840200238171304686441859603769017980249562449108238949606277054291}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{33} + \frac{17842233610987255577562206349723210902878560222016790298801786057975098601940981}{1032405358830173892910861790294858942399024693403321289153880613291863871074308619436032} a^{32} - \frac{904938298496867246491169585890572205240542813769849598247398512080983481065146433}{1032405358830173892910861790294858942399024693403321289153880613291863871074308619436032} a^{31} + \frac{4166762417482036565068913611094761127238889619426734310403782733817798407543892969}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{30} + \frac{6134274533165728810406706446645195373589686525510904823265648751283753720462774213}{1032405358830173892910861790294858942399024693403321289153880613291863871074308619436032} a^{29} - \frac{25660013553443246103099744978292027856660497352128823287890271372766717979507263937}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{28} + \frac{38412470270311869024643904561812767864922656359928428201939580491361190906194085889}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{27} + \frac{47835840425644954778744499330562414189324468860480455703102225034949077419187789433}{1032405358830173892910861790294858942399024693403321289153880613291863871074308619436032} a^{26} + \frac{1846514446420899355730608565091650363971560032327921702112970312171559607056833986741}{1032405358830173892910861790294858942399024693403321289153880613291863871074308619436032} a^{25} - \frac{91621743060827707189677727364347267428639607173239808519030842199465224758780014447}{18272661218233166246209943191059450307947339706253474144316471031714404797775373795328} a^{24} - \frac{14479857745375336999247561842418318553329859475591201738461352323471568319946376690937}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{23} + \frac{2234958771104879430778646486263767643462747077271364537655400729965294915808803130111}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{22} + \frac{37758615210350569676871410668750643168480180383535006102985274658001332096471205877333}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{21} - \frac{33053724757435358635580002938866075733787279093765812563438824350781001943940901723599}{1032405358830173892910861790294858942399024693403321289153880613291863871074308619436032} a^{20} + \frac{362996150989584819848213091067689261179478038521938356200017252673729662385236773610671}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{19} + \frac{6274309145026097530069126888899600265801108821294422249234367984771261342850351907185}{18272661218233166246209943191059450307947339706253474144316471031714404797775373795328} a^{18} - \frac{989846665067769514930733585520584655843070249261444749002114398044118427826066759736483}{2064810717660347785821723580589717884798049386806642578307761226583727742148617238872064} a^{17} + \frac{233149117830714350125097290971718729786548355951725077372988424824847614944014349974275}{516202679415086946455430895147429471199512346701660644576940306645931935537154309718016} a^{16} - \frac{201423948890139689629483227276937856444309091014023671792377498974764032507187014189437}{516202679415086946455430895147429471199512346701660644576940306645931935537154309718016} a^{15} - \frac{13857760540662641688456331622596222846259337138531138551864631731696264695914870182757}{64525334926885868306928861893428683899939043337707580572117538330741491942144288714752} a^{14} - \frac{14296674835445033479185920647033794901813997911765189985215328266268617026551291526927}{64525334926885868306928861893428683899939043337707580572117538330741491942144288714752} a^{13} + \frac{1896676666350727662977892082130819984882554115129451586486750447988680674682982901803}{64525334926885868306928861893428683899939043337707580572117538330741491942144288714752} a^{12} + \frac{336504465984212657072342276962980757676616205221125539963086724803729165688789412353}{2016416716465183384591526934169646371873095104303361892878673072835671623192009022336} a^{11} - \frac{421838646975163555193085868693602542386151801671474279233272157713485816696354654093}{4032833432930366769183053868339292743746190208606723785757346145671343246384018044672} a^{10} + \frac{4266877999885084436216721149881561638880521981956355518259235409918447702896124923}{54497749093653604988960187409990442483056624440631402510234407373937070897081324928} a^{9} + \frac{409197638425552846157305456771073522469641659751433160945465315592894633820987546419}{1008208358232591692295763467084823185936547552151680946439336536417835811596004511168} a^{8} + \frac{258191886339446130347623179400274844936703354336590697459028235330019411711905092711}{2016416716465183384591526934169646371873095104303361892878673072835671623192009022336} a^{7} + \frac{53978700270459680548817737941668093671833662414770865123930520277294668633383093467}{1008208358232591692295763467084823185936547552151680946439336536417835811596004511168} a^{6} - \frac{123754561690687949293982922607263082558584694837358229083060463618717720786166083349}{252052089558147923073940866771205796484136888037920236609834134104458952899001127792} a^{5} - \frac{23807323734830795768306494205947010143029527382454164801077534478463157184996569017}{126026044779073961536970433385602898242068444018960118304917067052229476449500563896} a^{4} + \frac{7429100606702445569158975802685552233034454887662574835840673264011061145821728069}{31506511194768490384242608346400724560517111004740029576229266763057369112375140974} a^{3} + \frac{5810056156114776380208192704520371885790758793192332260187715812478097877933566971}{31506511194768490384242608346400724560517111004740029576229266763057369112375140974} a^{2} - \frac{5179725975499813080615843448218657284885171124095164529831104869119430486441935101}{31506511194768490384242608346400724560517111004740029576229266763057369112375140974} a - \frac{6735147085749095445777939296208641523203193604211392478854204825835740101840082512}{15753255597384245192121304173200362280258555502370014788114633381528684556187570487}$

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

Class group and class number

$C_{2}\times C_{666}$, which has order $1332$ (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{6899910453458373647534320032271204564955814832839686294007683}{5950597513011444849186726852202315520707415818939582428264681832448} a^{35} - \frac{4254487377150127589073312665503649423247614369439731423655417}{5950597513011444849186726852202315520707415818939582428264681832448} a^{34} - \frac{174852457054768650356695803948864245787037338652787798351490101}{5950597513011444849186726852202315520707415818939582428264681832448} a^{33} + \frac{8306003440207820692753610823319844132665569652445026913316469}{2975298756505722424593363426101157760353707909469791214132340916224} a^{32} + \frac{1178202165471456096202169685501653279419327161202797908454208447}{2975298756505722424593363426101157760353707909469791214132340916224} a^{31} + \frac{650263574297343635135955789810182424161627386113822143434491195}{5950597513011444849186726852202315520707415818939582428264681832448} a^{30} - \frac{12401777535089300772761683742298283746406531838826852393663407897}{2975298756505722424593363426101157760353707909469791214132340916224} a^{29} - \frac{8927179419424481982964149158418673898599240188645030517981814359}{5950597513011444849186726852202315520707415818939582428264681832448} a^{28} + \frac{217174954291347054267695040401525349099609491924782526906813981755}{5950597513011444849186726852202315520707415818939582428264681832448} a^{27} + \frac{43835504115090200680999289185865219954106119298623052303086436809}{2975298756505722424593363426101157760353707909469791214132340916224} a^{26} - \frac{757097374684012834560007725647790739459627519229503410638453883487}{2975298756505722424593363426101157760353707909469791214132340916224} a^{25} - \frac{811534722041352071021199872009105686843450920388329589270355305981}{5950597513011444849186726852202315520707415818939582428264681832448} a^{24} + \frac{8686313426879199633142511345890879576431351471641116650481824254677}{5950597513011444849186726852202315520707415818939582428264681832448} a^{23} + \frac{5128086331765098520807916528396993764175464408581754528564632491649}{5950597513011444849186726852202315520707415818939582428264681832448} a^{22} - \frac{42464040243473795797326737371533099246017244923108012080388520859661}{5950597513011444849186726852202315520707415818939582428264681832448} a^{21} - \frac{11728594789346376475741651355198056053445629992758252322237494302291}{2975298756505722424593363426101157760353707909469791214132340916224} a^{20} + \frac{171311872357082161263798835198797653098045159829325339531481544910913}{5950597513011444849186726852202315520707415818939582428264681832448} a^{19} + \frac{100462210540939635504607047385565926458484283394405765217477636682555}{5950597513011444849186726852202315520707415818939582428264681832448} a^{18} - \frac{549981358919474744805010945801325118824071686285882651670199623152269}{5950597513011444849186726852202315520707415818939582428264681832448} a^{17} - \frac{22408191518735246163897169424795306092895673388848818097856271774675}{371912344563215303074170428262644720044213488683723901766542614528} a^{16} + \frac{4851405337992890884318323526437733489049372166158044783674536411379}{20103369976390016382387590716899714596984512901822913609002303488} a^{15} + \frac{101104469134511414379784996368247515211965197671467207210960184738425}{743824689126430606148340856525289440088426977367447803533085229056} a^{14} - \frac{189003138819552396835204196893407816616286548886599070759716303387761}{371912344563215303074170428262644720044213488683723901766542614528} a^{13} - \frac{41778604517509827130760692138526797477962246987620608758405686770817}{185956172281607651537085214131322360022106744341861950883271307264} a^{12} + \frac{74260673396896712558797780419560141289560351146138216168619401184139}{92978086140803825768542607065661180011053372170930975441635653632} a^{11} + \frac{19717297244574060670524364806019441440935577095524160570947407579329}{46489043070401912884271303532830590005526686085465487720817826816} a^{10} - \frac{9629883360220375170812499114309244249694461207324825035048149750981}{11622260767600478221067825883207647501381671521366371930204456704} a^{9} - \frac{6738375529667967079935450839340779620600634770420795267131975228255}{11622260767600478221067825883207647501381671521366371930204456704} a^{8} + \frac{434617206573991762237897876752057056068294660390773621399032383173}{726391297975029888816739117700477968836354470085398245637778544} a^{7} + \frac{841159000364202722192327259736612164995048998815916873442956362585}{2905565191900119555266956470801911875345417880341592982551114176} a^{6} - \frac{110187908899682664474348266154301191045761142435599708505465070571}{363195648987514944408369558850238984418177235042699122818889272} a^{5} - \frac{853918721011941032730626902261697365109281332140812688993081391}{45399456123439368051046194856279873052272154380337390352361159} a^{4} + \frac{22310427920902764234028824771936090170558897589152224112474700523}{181597824493757472204184779425119492209088617521349561409444636} a^{3} + \frac{4113842456258695327193951042495455086601434581973420683962748365}{181597824493757472204184779425119492209088617521349561409444636} a^{2} - \frac{304401204619835715683048089508136338741620451141632209057431489}{90798912246878736102092389712559746104544308760674780704722318} a - \frac{51160113881630881169349542598133410735222574289887138232917719}{45399456123439368051046194856279873052272154380337390352361159} \) (order $6$)
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:  \( 1513299055779282.5 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), 3.3.361.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.0.3518667.1, 6.6.1206902781.1, 6.0.44700103.1, \(\Q(\zeta_{19})^+\), 12.0.1456614322785533961.1, 18.0.5677392343251487443465123.1, 18.18.229103259404379626459026291748661.1, 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 $18^{2}$ R $18^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{36}$ $18^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$ $18^{2}$

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
3Data not computed
$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