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Properties

Label 18.0.292...248.2
Degree $18$
Signature $[0, 9]$
Discriminant $-2.925\times 10^{44}$
Root discriminant \(295.35\)
Ramified primes $2,3,19$
Class number $18962019$ (GRH)
Class group [3, 3, 3, 111, 6327] (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 324*x^16 + 44775*x^14 - 3393108*x^12 + 151870707*x^10 - 2236034*x^9 - 4032329868*x^8 + 543356262*x^7 + 60647896506*x^6 - 38980780722*x^5 - 456682273992*x^4 + 951034452948*x^3 + 1449222098553*x^2 - 5840898697746*x + 6529918591491)
 
gp: K = bnfinit(y^18 - 324*y^16 + 44775*y^14 - 3393108*y^12 + 151870707*y^10 - 2236034*y^9 - 4032329868*y^8 + 543356262*y^7 + 60647896506*y^6 - 38980780722*y^5 - 456682273992*y^4 + 951034452948*y^3 + 1449222098553*y^2 - 5840898697746*y + 6529918591491, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 324*x^16 + 44775*x^14 - 3393108*x^12 + 151870707*x^10 - 2236034*x^9 - 4032329868*x^8 + 543356262*x^7 + 60647896506*x^6 - 38980780722*x^5 - 456682273992*x^4 + 951034452948*x^3 + 1449222098553*x^2 - 5840898697746*x + 6529918591491);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 324*x^16 + 44775*x^14 - 3393108*x^12 + 151870707*x^10 - 2236034*x^9 - 4032329868*x^8 + 543356262*x^7 + 60647896506*x^6 - 38980780722*x^5 - 456682273992*x^4 + 951034452948*x^3 + 1449222098553*x^2 - 5840898697746*x + 6529918591491)
 

\( x^{18} - 324 x^{16} + 44775 x^{14} - 3393108 x^{12} + 151870707 x^{10} - 2236034 x^{9} + \cdots + 6529918591491 \) Copy content Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-292542051093996550620636964156035982581301248\) \(\medspace = -\,2^{27}\cdot 3^{44}\cdot 19^{12}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(295.35\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}3^{22/9}19^{2/3}\approx 295.3546376705848$
Ramified primes:   \(2\), \(3\), \(19\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-2}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4104=2^{3}\cdot 3^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4104}(1,·)$, $\chi_{4104}(1027,·)$, $\chi_{4104}(961,·)$, $\chi_{4104}(1489,·)$, $\chi_{4104}(2515,·)$, $\chi_{4104}(2329,·)$, $\chi_{4104}(3355,·)$, $\chi_{4104}(1369,·)$, $\chi_{4104}(2395,·)$, $\chi_{4104}(1987,·)$, $\chi_{4104}(619,·)$, $\chi_{4104}(3697,·)$, $\chi_{4104}(2857,·)$, $\chi_{4104}(3883,·)$, $\chi_{4104}(2737,·)$, $\chi_{4104}(3763,·)$, $\chi_{4104}(121,·)$, $\chi_{4104}(1147,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{256}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{19}a^{6}+\frac{6}{19}a^{4}-\frac{7}{19}a^{2}+\frac{8}{19}$, $\frac{1}{19}a^{7}+\frac{6}{19}a^{5}-\frac{7}{19}a^{3}+\frac{8}{19}a$, $\frac{1}{19}a^{8}-\frac{5}{19}a^{4}-\frac{7}{19}a^{2}+\frac{9}{19}$, $\frac{1}{323}a^{9}-\frac{6}{323}a^{8}-\frac{2}{323}a^{7}+\frac{1}{323}a^{6}-\frac{36}{323}a^{5}+\frac{55}{323}a^{4}-\frac{31}{323}a^{3}-\frac{22}{323}a^{2}-\frac{159}{323}a+\frac{4}{19}$, $\frac{1}{323}a^{10}-\frac{4}{323}a^{8}+\frac{6}{323}a^{7}+\frac{4}{323}a^{6}-\frac{59}{323}a^{5}+\frac{10}{323}a^{4}-\frac{4}{323}a^{3}-\frac{121}{323}a^{2}-\frac{104}{323}a+\frac{1}{19}$, $\frac{1}{323}a^{11}-\frac{1}{323}a^{8}-\frac{4}{323}a^{7}-\frac{4}{323}a^{6}-\frac{134}{323}a^{5}+\frac{6}{17}a^{4}+\frac{78}{323}a^{3}-\frac{22}{323}a^{2}+\frac{27}{323}a-\frac{8}{19}$, $\frac{1}{6137}a^{12}-\frac{7}{6137}a^{10}-\frac{16}{6137}a^{8}-\frac{7}{323}a^{7}-\frac{144}{6137}a^{6}-\frac{137}{323}a^{5}-\frac{1280}{6137}a^{4}+\frac{30}{323}a^{3}+\frac{2263}{6137}a^{2}-\frac{132}{323}a+\frac{158}{361}$, $\frac{1}{6137}a^{13}-\frac{7}{6137}a^{11}+\frac{3}{6137}a^{9}+\frac{4}{323}a^{8}+\frac{141}{6137}a^{7}-\frac{26}{6137}a^{5}-\frac{9}{19}a^{4}-\frac{587}{6137}a^{3}+\frac{67}{323}a^{2}+\frac{2249}{6137}a+\frac{1}{19}$, $\frac{1}{6137}a^{14}-\frac{8}{6137}a^{10}+\frac{10}{6137}a^{8}+\frac{5}{323}a^{7}+\frac{11}{6137}a^{6}+\frac{87}{323}a^{5}+\frac{219}{6137}a^{4}+\frac{155}{323}a^{3}-\frac{96}{361}a^{2}+\frac{116}{323}a+\frac{42}{361}$, $\frac{1}{6137}a^{15}-\frac{8}{6137}a^{11}-\frac{9}{6137}a^{9}-\frac{6}{323}a^{8}+\frac{49}{6137}a^{7}+\frac{1}{323}a^{6}+\frac{903}{6137}a^{5}-\frac{2}{323}a^{4}-\frac{1043}{6137}a^{3}-\frac{117}{323}a^{2}-\frac{2402}{6137}a+\frac{4}{19}$, $\frac{1}{104329}a^{16}-\frac{2}{104329}a^{15}+\frac{2}{104329}a^{14}-\frac{2}{104329}a^{13}-\frac{1}{104329}a^{12}+\frac{106}{104329}a^{11}-\frac{1}{6137}a^{10}-\frac{140}{104329}a^{9}+\frac{926}{104329}a^{8}-\frac{69}{5491}a^{7}-\frac{235}{104329}a^{6}-\frac{43098}{104329}a^{5}-\frac{44335}{104329}a^{4}-\frac{29040}{104329}a^{3}-\frac{20852}{104329}a^{2}+\frac{18}{6137}a-\frac{101}{361}$, $\frac{1}{13\!\cdots\!77}a^{17}-\frac{78\!\cdots\!02}{13\!\cdots\!77}a^{16}+\frac{45\!\cdots\!75}{81\!\cdots\!81}a^{15}-\frac{49\!\cdots\!72}{81\!\cdots\!81}a^{14}-\frac{92\!\cdots\!84}{13\!\cdots\!77}a^{13}-\frac{71\!\cdots\!75}{13\!\cdots\!77}a^{12}+\frac{89\!\cdots\!97}{72\!\cdots\!83}a^{11}-\frac{46\!\cdots\!98}{13\!\cdots\!77}a^{10}+\frac{17\!\cdots\!06}{13\!\cdots\!77}a^{9}+\frac{19\!\cdots\!85}{13\!\cdots\!77}a^{8}+\frac{68\!\cdots\!00}{13\!\cdots\!77}a^{7}+\frac{18\!\cdots\!13}{81\!\cdots\!81}a^{6}-\frac{57\!\cdots\!83}{13\!\cdots\!77}a^{5}+\frac{23\!\cdots\!12}{13\!\cdots\!77}a^{4}+\frac{45\!\cdots\!68}{13\!\cdots\!77}a^{3}-\frac{32\!\cdots\!50}{13\!\cdots\!77}a^{2}+\frac{37\!\cdots\!63}{81\!\cdots\!81}a-\frac{47\!\cdots\!62}{47\!\cdots\!93}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $17$

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{111}\times C_{6327}$, which has order $18962019$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{51\!\cdots\!20}{25\!\cdots\!23}a^{17}+\frac{21\!\cdots\!00}{28\!\cdots\!61}a^{16}-\frac{17\!\cdots\!02}{25\!\cdots\!23}a^{15}-\frac{19\!\cdots\!36}{47\!\cdots\!37}a^{14}+\frac{24\!\cdots\!62}{25\!\cdots\!23}a^{13}+\frac{38\!\cdots\!80}{47\!\cdots\!37}a^{12}-\frac{18\!\cdots\!68}{25\!\cdots\!23}a^{11}-\frac{37\!\cdots\!92}{47\!\cdots\!37}a^{10}+\frac{49\!\cdots\!22}{14\!\cdots\!19}a^{9}+\frac{20\!\cdots\!56}{47\!\cdots\!37}a^{8}-\frac{22\!\cdots\!84}{25\!\cdots\!23}a^{7}-\frac{58\!\cdots\!93}{47\!\cdots\!37}a^{6}+\frac{31\!\cdots\!56}{25\!\cdots\!23}a^{5}+\frac{67\!\cdots\!44}{47\!\cdots\!37}a^{4}-\frac{22\!\cdots\!54}{25\!\cdots\!23}a^{3}+\frac{98\!\cdots\!95}{47\!\cdots\!37}a^{2}+\frac{55\!\cdots\!00}{14\!\cdots\!19}a-\frac{79\!\cdots\!32}{16\!\cdots\!33}$, $\frac{42\!\cdots\!20}{42\!\cdots\!91}a^{17}+\frac{26\!\cdots\!96}{81\!\cdots\!29}a^{16}-\frac{48\!\cdots\!62}{14\!\cdots\!19}a^{15}-\frac{58\!\cdots\!96}{28\!\cdots\!61}a^{14}+\frac{19\!\cdots\!34}{42\!\cdots\!91}a^{13}+\frac{35\!\cdots\!39}{81\!\cdots\!29}a^{12}-\frac{14\!\cdots\!84}{42\!\cdots\!91}a^{11}-\frac{36\!\cdots\!88}{81\!\cdots\!29}a^{10}+\frac{67\!\cdots\!22}{42\!\cdots\!91}a^{9}+\frac{20\!\cdots\!90}{81\!\cdots\!29}a^{8}-\frac{17\!\cdots\!00}{42\!\cdots\!91}a^{7}-\frac{35\!\cdots\!31}{47\!\cdots\!37}a^{6}+\frac{25\!\cdots\!28}{42\!\cdots\!91}a^{5}+\frac{77\!\cdots\!63}{81\!\cdots\!29}a^{4}-\frac{18\!\cdots\!98}{42\!\cdots\!91}a^{3}-\frac{14\!\cdots\!35}{81\!\cdots\!29}a^{2}+\frac{62\!\cdots\!24}{25\!\cdots\!23}a-\frac{73\!\cdots\!43}{28\!\cdots\!61}$, $\frac{56\!\cdots\!32}{13\!\cdots\!77}a^{17}-\frac{37\!\cdots\!93}{13\!\cdots\!77}a^{16}-\frac{11\!\cdots\!18}{42\!\cdots\!99}a^{15}+\frac{69\!\cdots\!53}{81\!\cdots\!81}a^{14}-\frac{19\!\cdots\!92}{13\!\cdots\!77}a^{13}-\frac{15\!\cdots\!55}{13\!\cdots\!77}a^{12}+\frac{37\!\cdots\!02}{13\!\cdots\!77}a^{11}+\frac{11\!\cdots\!15}{13\!\cdots\!77}a^{10}-\frac{30\!\cdots\!04}{13\!\cdots\!77}a^{9}-\frac{12\!\cdots\!79}{38\!\cdots\!57}a^{8}+\frac{12\!\cdots\!92}{13\!\cdots\!77}a^{7}+\frac{59\!\cdots\!23}{81\!\cdots\!81}a^{6}-\frac{26\!\cdots\!32}{13\!\cdots\!77}a^{5}-\frac{11\!\cdots\!08}{13\!\cdots\!77}a^{4}+\frac{31\!\cdots\!84}{13\!\cdots\!77}a^{3}+\frac{28\!\cdots\!70}{13\!\cdots\!77}a^{2}-\frac{11\!\cdots\!02}{81\!\cdots\!81}a+\frac{79\!\cdots\!88}{47\!\cdots\!93}$, $\frac{53\!\cdots\!66}{13\!\cdots\!77}a^{17}-\frac{36\!\cdots\!17}{13\!\cdots\!77}a^{16}-\frac{51\!\cdots\!84}{42\!\cdots\!99}a^{15}+\frac{63\!\cdots\!01}{81\!\cdots\!81}a^{14}+\frac{21\!\cdots\!06}{13\!\cdots\!77}a^{13}-\frac{70\!\cdots\!03}{72\!\cdots\!83}a^{12}-\frac{14\!\cdots\!88}{13\!\cdots\!77}a^{11}+\frac{88\!\cdots\!46}{13\!\cdots\!77}a^{10}+\frac{52\!\cdots\!20}{13\!\cdots\!77}a^{9}-\frac{32\!\cdots\!24}{13\!\cdots\!77}a^{8}-\frac{95\!\cdots\!06}{13\!\cdots\!77}a^{7}+\frac{37\!\cdots\!98}{81\!\cdots\!81}a^{6}+\frac{51\!\cdots\!56}{13\!\cdots\!77}a^{5}-\frac{65\!\cdots\!05}{13\!\cdots\!77}a^{4}+\frac{80\!\cdots\!50}{13\!\cdots\!77}a^{3}+\frac{24\!\cdots\!16}{13\!\cdots\!77}a^{2}-\frac{41\!\cdots\!32}{81\!\cdots\!81}a+\frac{24\!\cdots\!88}{47\!\cdots\!93}$, $\frac{11\!\cdots\!62}{81\!\cdots\!81}a^{17}+\frac{46\!\cdots\!69}{81\!\cdots\!81}a^{16}-\frac{18\!\cdots\!52}{42\!\cdots\!99}a^{15}-\frac{13\!\cdots\!09}{81\!\cdots\!81}a^{14}+\frac{46\!\cdots\!82}{81\!\cdots\!81}a^{13}+\frac{16\!\cdots\!96}{81\!\cdots\!81}a^{12}-\frac{33\!\cdots\!26}{81\!\cdots\!81}a^{11}-\frac{10\!\cdots\!58}{81\!\cdots\!81}a^{10}+\frac{14\!\cdots\!02}{81\!\cdots\!81}a^{9}+\frac{24\!\cdots\!56}{47\!\cdots\!93}a^{8}-\frac{38\!\cdots\!60}{81\!\cdots\!81}a^{7}-\frac{94\!\cdots\!72}{81\!\cdots\!81}a^{6}+\frac{58\!\cdots\!24}{81\!\cdots\!81}a^{5}+\frac{10\!\cdots\!89}{81\!\cdots\!81}a^{4}-\frac{46\!\cdots\!48}{81\!\cdots\!81}a^{3}-\frac{12\!\cdots\!67}{81\!\cdots\!81}a^{2}+\frac{13\!\cdots\!36}{47\!\cdots\!93}a-\frac{61\!\cdots\!76}{28\!\cdots\!29}$, $\frac{61\!\cdots\!02}{13\!\cdots\!77}a^{17}+\frac{12\!\cdots\!37}{13\!\cdots\!77}a^{16}-\frac{61\!\cdots\!04}{42\!\cdots\!99}a^{15}-\frac{22\!\cdots\!87}{81\!\cdots\!81}a^{14}+\frac{27\!\cdots\!02}{13\!\cdots\!77}a^{13}+\frac{53\!\cdots\!72}{13\!\cdots\!77}a^{12}-\frac{20\!\cdots\!32}{13\!\cdots\!77}a^{11}-\frac{39\!\cdots\!34}{13\!\cdots\!77}a^{10}+\frac{92\!\cdots\!28}{13\!\cdots\!77}a^{9}+\frac{17\!\cdots\!67}{13\!\cdots\!77}a^{8}-\frac{24\!\cdots\!54}{13\!\cdots\!77}a^{7}-\frac{26\!\cdots\!10}{81\!\cdots\!81}a^{6}+\frac{34\!\cdots\!76}{13\!\cdots\!77}a^{5}+\frac{50\!\cdots\!64}{13\!\cdots\!77}a^{4}-\frac{26\!\cdots\!22}{13\!\cdots\!77}a^{3}-\frac{28\!\cdots\!80}{13\!\cdots\!77}a^{2}+\frac{74\!\cdots\!28}{81\!\cdots\!81}a-\frac{35\!\cdots\!80}{47\!\cdots\!93}$, $\frac{31\!\cdots\!42}{13\!\cdots\!77}a^{17}-\frac{15\!\cdots\!44}{13\!\cdots\!77}a^{16}-\frac{29\!\cdots\!18}{42\!\cdots\!99}a^{15}+\frac{27\!\cdots\!31}{81\!\cdots\!81}a^{14}+\frac{12\!\cdots\!84}{13\!\cdots\!77}a^{13}-\frac{60\!\cdots\!87}{13\!\cdots\!77}a^{12}-\frac{81\!\cdots\!32}{13\!\cdots\!77}a^{11}+\frac{41\!\cdots\!48}{13\!\cdots\!77}a^{10}+\frac{29\!\cdots\!50}{13\!\cdots\!77}a^{9}-\frac{16\!\cdots\!35}{13\!\cdots\!77}a^{8}-\frac{54\!\cdots\!54}{13\!\cdots\!77}a^{7}+\frac{19\!\cdots\!09}{81\!\cdots\!81}a^{6}+\frac{30\!\cdots\!36}{13\!\cdots\!77}a^{5}-\frac{34\!\cdots\!13}{13\!\cdots\!77}a^{4}+\frac{41\!\cdots\!80}{13\!\cdots\!77}a^{3}+\frac{67\!\cdots\!03}{72\!\cdots\!83}a^{2}-\frac{22\!\cdots\!32}{81\!\cdots\!81}a+\frac{11\!\cdots\!04}{47\!\cdots\!93}$, $\frac{13\!\cdots\!26}{13\!\cdots\!77}a^{17}+\frac{62\!\cdots\!15}{13\!\cdots\!77}a^{16}-\frac{12\!\cdots\!42}{42\!\cdots\!99}a^{15}-\frac{13\!\cdots\!83}{81\!\cdots\!81}a^{14}+\frac{56\!\cdots\!96}{13\!\cdots\!77}a^{13}+\frac{35\!\cdots\!95}{13\!\cdots\!77}a^{12}-\frac{41\!\cdots\!80}{13\!\cdots\!77}a^{11}-\frac{31\!\cdots\!78}{13\!\cdots\!77}a^{10}+\frac{18\!\cdots\!30}{13\!\cdots\!77}a^{9}+\frac{16\!\cdots\!58}{13\!\cdots\!77}a^{8}-\frac{45\!\cdots\!10}{13\!\cdots\!77}a^{7}-\frac{28\!\cdots\!55}{81\!\cdots\!81}a^{6}+\frac{61\!\cdots\!04}{13\!\cdots\!77}a^{5}+\frac{56\!\cdots\!60}{13\!\cdots\!77}a^{4}-\frac{41\!\cdots\!20}{13\!\cdots\!77}a^{3}+\frac{93\!\cdots\!71}{13\!\cdots\!77}a^{2}+\frac{10\!\cdots\!04}{81\!\cdots\!81}a-\frac{46\!\cdots\!22}{47\!\cdots\!93}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 200739576.95029396 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 200739576.95029396 \cdot 18962019}{2\cdot\sqrt{292542051093996550620636964156035982581301248}}\cr\approx \mathstrut & 1.69829155692178 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 324*x^16 + 44775*x^14 - 3393108*x^12 + 151870707*x^10 - 2236034*x^9 - 4032329868*x^8 + 543356262*x^7 + 60647896506*x^6 - 38980780722*x^5 - 456682273992*x^4 + 951034452948*x^3 + 1449222098553*x^2 - 5840898697746*x + 6529918591491)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^18 - 324*x^16 + 44775*x^14 - 3393108*x^12 + 151870707*x^10 - 2236034*x^9 - 4032329868*x^8 + 543356262*x^7 + 60647896506*x^6 - 38980780722*x^5 - 456682273992*x^4 + 951034452948*x^3 + 1449222098553*x^2 - 5840898697746*x + 6529918591491, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 324*x^16 + 44775*x^14 - 3393108*x^12 + 151870707*x^10 - 2236034*x^9 - 4032329868*x^8 + 543356262*x^7 + 60647896506*x^6 - 38980780722*x^5 - 456682273992*x^4 + 951034452948*x^3 + 1449222098553*x^2 - 5840898697746*x + 6529918591491);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 324*x^16 + 44775*x^14 - 3393108*x^12 + 151870707*x^10 - 2236034*x^9 - 4032329868*x^8 + 543356262*x^7 + 60647896506*x^6 - 38980780722*x^5 - 456682273992*x^4 + 951034452948*x^3 + 1449222098553*x^2 - 5840898697746*x + 6529918591491);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_{18}$ (as 18T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\zeta_{9})^+\), 6.0.3359232.1, 9.9.1476349596018920529.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R $18$ $18$ ${\href{/padicField/11.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/17.1.0.1}{1} }^{18}$ R $18$ $18$ $18$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.18.27.117$x^{18} + 16 x^{17} + 178 x^{16} + 2960 x^{15} + 45360 x^{14} + 447008 x^{13} + 3255456 x^{12} + 17891904 x^{11} + 60260960 x^{10} + 85138048 x^{9} - 288700480 x^{8} - 3555798272 x^{7} - 16235478272 x^{6} - 53744921088 x^{5} - 137665523200 x^{4} - 262308385792 x^{3} - 401534975744 x^{2} - 426755266560 x - 200836357632$$2$$9$$27$$C_{18}$$[3]^{9}$
\(3\) Copy content Toggle raw display 3.9.22.6$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.6$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$$9$$1$$22$$C_9$$[2, 3]$
\(19\) Copy content Toggle raw display 19.3.2.3$x^{3} + 38$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} + 38$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} + 38$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} + 38$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} + 38$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} + 38$$3$$1$$2$$C_3$$[\ ]_{3}$