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Properties

Label 18.0.242...096.1
Degree $18$
Signature $[0, 9]$
Discriminant $-2.425\times 10^{50}$
Root discriminant \(629.72\)
Ramified primes $2,3,7,17$
Class number $324$ (GRH)
Class group [3, 3, 3, 12] (GRH)
Galois group $C_3^2:D_6$ (as 18T52)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 237*x^16 - 896*x^15 + 28503*x^14 - 127218*x^13 + 2544555*x^12 - 8429568*x^11 + 155383248*x^10 - 475665960*x^9 + 7557303396*x^8 - 26162306688*x^7 + 232899383856*x^6 - 1038222554208*x^5 + 6045973402320*x^4 - 34008769056256*x^3 + 141085549266816*x^2 - 513307676719488*x + 1004539113344704)
 
gp: K = bnfinit(y^18 - 6*y^17 + 237*y^16 - 896*y^15 + 28503*y^14 - 127218*y^13 + 2544555*y^12 - 8429568*y^11 + 155383248*y^10 - 475665960*y^9 + 7557303396*y^8 - 26162306688*y^7 + 232899383856*y^6 - 1038222554208*y^5 + 6045973402320*y^4 - 34008769056256*y^3 + 141085549266816*y^2 - 513307676719488*y + 1004539113344704, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 237*x^16 - 896*x^15 + 28503*x^14 - 127218*x^13 + 2544555*x^12 - 8429568*x^11 + 155383248*x^10 - 475665960*x^9 + 7557303396*x^8 - 26162306688*x^7 + 232899383856*x^6 - 1038222554208*x^5 + 6045973402320*x^4 - 34008769056256*x^3 + 141085549266816*x^2 - 513307676719488*x + 1004539113344704);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 237*x^16 - 896*x^15 + 28503*x^14 - 127218*x^13 + 2544555*x^12 - 8429568*x^11 + 155383248*x^10 - 475665960*x^9 + 7557303396*x^8 - 26162306688*x^7 + 232899383856*x^6 - 1038222554208*x^5 + 6045973402320*x^4 - 34008769056256*x^3 + 141085549266816*x^2 - 513307676719488*x + 1004539113344704)
 

\( x^{18} - 6 x^{17} + 237 x^{16} - 896 x^{15} + 28503 x^{14} - 127218 x^{13} + 2544555 x^{12} + \cdots + 10\!\cdots\!04 \) 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:   \(-242459919475028479855663400956723947300369356292096\) \(\medspace = -\,2^{33}\cdot 3^{30}\cdot 7^{12}\cdot 17^{13}\) 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:  \(629.72\)
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:  $2^{11/6}3^{11/6}7^{2/3}17^{5/6}\approx 1036.056453367024$
Ramified primes:   \(2\), \(3\), \(7\), \(17\) 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:  \(\Q(\sqrt{-34}) \)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{32}a^{10}-\frac{1}{16}a^{9}-\frac{3}{32}a^{8}-\frac{1}{8}a^{7}+\frac{7}{32}a^{6}+\frac{1}{16}a^{5}-\frac{13}{32}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a+\frac{3}{8}$, $\frac{1}{32}a^{11}+\frac{1}{32}a^{9}-\frac{1}{16}a^{8}+\frac{7}{32}a^{7}-\frac{1}{4}a^{6}-\frac{1}{32}a^{5}-\frac{7}{16}a^{4}+\frac{1}{4}a^{2}-\frac{1}{8}a+\frac{1}{4}$, $\frac{1}{91392}a^{12}+\frac{13}{15232}a^{11}+\frac{409}{30464}a^{10}-\frac{67}{1428}a^{9}-\frac{529}{30464}a^{8}+\frac{2087}{15232}a^{7}+\frac{49}{4352}a^{6}-\frac{1}{112}a^{5}+\frac{6151}{15232}a^{4}+\frac{3109}{11424}a^{3}-\frac{193}{1088}a^{2}-\frac{45}{136}a-\frac{725}{1632}$, $\frac{1}{9778944}a^{13}+\frac{1}{9778944}a^{12}+\frac{311}{3259648}a^{11}-\frac{30223}{9778944}a^{10}+\frac{265757}{9778944}a^{9}-\frac{4597}{3259648}a^{8}+\frac{98567}{465664}a^{7}+\frac{34245}{3259648}a^{6}-\frac{178537}{1629824}a^{5}+\frac{2161555}{4889472}a^{4}-\frac{32257}{349248}a^{3}+\frac{2853}{6848}a^{2}-\frac{48497}{174624}a+\frac{68881}{174624}$, $\frac{1}{156463104}a^{14}+\frac{1}{78231552}a^{13}-\frac{671}{156463104}a^{12}+\frac{223}{76398}a^{11}+\frac{1933303}{156463104}a^{10}-\frac{4683881}{78231552}a^{9}-\frac{4984879}{52154368}a^{8}+\frac{214369}{1629824}a^{7}+\frac{112771}{1629824}a^{6}+\frac{1622063}{9778944}a^{5}+\frac{6469271}{19557888}a^{4}+\frac{94891}{1222368}a^{3}+\frac{484559}{1396992}a^{2}+\frac{254513}{698496}a+\frac{136037}{1396992}$, $\frac{1}{156463104}a^{15}-\frac{1}{52154368}a^{13}-\frac{7}{11175936}a^{12}+\frac{1677}{3067904}a^{11}+\frac{43}{1629824}a^{10}-\frac{6484553}{156463104}a^{9}+\frac{1100751}{26077184}a^{8}+\frac{108151}{814912}a^{7}+\frac{1356989}{9778944}a^{6}-\frac{165497}{931328}a^{5}+\frac{438695}{3259648}a^{4}-\frac{807733}{3259648}a^{3}+\frac{24337}{116416}a^{2}-\frac{113733}{465664}a+\frac{299411}{698496}$, $\frac{1}{5625161515008}a^{16}-\frac{1081}{2812580757504}a^{15}+\frac{1721}{803594502144}a^{14}-\frac{16393}{703145189376}a^{13}+\frac{60173}{17523867648}a^{12}+\frac{12883347425}{2812580757504}a^{11}-\frac{698021185}{52571602944}a^{10}+\frac{37563081655}{703145189376}a^{9}+\frac{23721754617}{312508973056}a^{8}+\frac{53616676355}{351572594688}a^{7}+\frac{38943607111}{703145189376}a^{6}+\frac{8843389423}{87893148672}a^{5}+\frac{27170411785}{58595432448}a^{4}+\frac{58148591909}{175786297344}a^{3}-\frac{13409166425}{50224656384}a^{2}-\frac{484697779}{6278082048}a-\frac{6482265161}{25112328192}$, $\frac{1}{41\!\cdots\!12}a^{17}+\frac{30\!\cdots\!97}{41\!\cdots\!12}a^{16}-\frac{15\!\cdots\!41}{13\!\cdots\!04}a^{15}+\frac{29\!\cdots\!51}{13\!\cdots\!04}a^{14}-\frac{16\!\cdots\!63}{41\!\cdots\!12}a^{13}+\frac{76\!\cdots\!33}{41\!\cdots\!12}a^{12}-\frac{32\!\cdots\!77}{41\!\cdots\!12}a^{11}+\frac{76\!\cdots\!29}{13\!\cdots\!04}a^{10}-\frac{16\!\cdots\!25}{29\!\cdots\!08}a^{9}-\frac{18\!\cdots\!49}{20\!\cdots\!56}a^{8}-\frac{10\!\cdots\!31}{52\!\cdots\!64}a^{7}+\frac{11\!\cdots\!19}{17\!\cdots\!88}a^{6}-\frac{53\!\cdots\!03}{13\!\cdots\!16}a^{5}+\frac{27\!\cdots\!65}{54\!\cdots\!32}a^{4}-\frac{69\!\cdots\!65}{15\!\cdots\!96}a^{3}-\frac{74\!\cdots\!51}{37\!\cdots\!76}a^{2}-\frac{84\!\cdots\!67}{61\!\cdots\!96}a+\frac{12\!\cdots\!09}{10\!\cdots\!64}$ 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:  $2$

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{12}$, which has order $324$ (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:   $\frac{60\!\cdots\!93}{43\!\cdots\!28}a^{17}-\frac{48\!\cdots\!89}{87\!\cdots\!56}a^{16}+\frac{16\!\cdots\!95}{54\!\cdots\!16}a^{15}-\frac{55\!\cdots\!19}{87\!\cdots\!56}a^{14}+\frac{53\!\cdots\!07}{14\!\cdots\!76}a^{13}-\frac{13\!\cdots\!63}{12\!\cdots\!08}a^{12}+\frac{33\!\cdots\!01}{10\!\cdots\!32}a^{11}-\frac{46\!\cdots\!25}{87\!\cdots\!56}a^{10}+\frac{13\!\cdots\!23}{72\!\cdots\!88}a^{9}-\frac{12\!\cdots\!89}{43\!\cdots\!28}a^{8}+\frac{60\!\cdots\!51}{68\!\cdots\!52}a^{7}-\frac{27\!\cdots\!97}{15\!\cdots\!76}a^{6}+\frac{11\!\cdots\!35}{45\!\cdots\!68}a^{5}-\frac{21\!\cdots\!41}{27\!\cdots\!08}a^{4}+\frac{41\!\cdots\!73}{68\!\cdots\!52}a^{3}-\frac{19\!\cdots\!11}{77\!\cdots\!88}a^{2}+\frac{22\!\cdots\!03}{19\!\cdots\!72}a-\frac{34\!\cdots\!61}{12\!\cdots\!48}$, $\frac{75\!\cdots\!41}{10\!\cdots\!28}a^{17}+\frac{72\!\cdots\!81}{20\!\cdots\!56}a^{16}+\frac{13\!\cdots\!69}{74\!\cdots\!52}a^{15}+\frac{29\!\cdots\!99}{20\!\cdots\!56}a^{14}+\frac{80\!\cdots\!39}{34\!\cdots\!76}a^{13}-\frac{50\!\cdots\!71}{20\!\cdots\!56}a^{12}+\frac{81\!\cdots\!55}{52\!\cdots\!64}a^{11}-\frac{10\!\cdots\!31}{20\!\cdots\!56}a^{10}+\frac{11\!\cdots\!95}{17\!\cdots\!88}a^{9}-\frac{40\!\cdots\!95}{10\!\cdots\!28}a^{8}+\frac{15\!\cdots\!75}{65\!\cdots\!08}a^{7}-\frac{40\!\cdots\!21}{26\!\cdots\!32}a^{6}+\frac{88\!\cdots\!91}{10\!\cdots\!68}a^{5}-\frac{32\!\cdots\!55}{65\!\cdots\!08}a^{4}+\frac{11\!\cdots\!89}{46\!\cdots\!72}a^{3}-\frac{20\!\cdots\!81}{18\!\cdots\!88}a^{2}+\frac{15\!\cdots\!91}{46\!\cdots\!72}a-\frac{27\!\cdots\!21}{44\!\cdots\!64}$, $\frac{62\!\cdots\!21}{10\!\cdots\!28}a^{17}-\frac{12\!\cdots\!95}{20\!\cdots\!56}a^{16}+\frac{31\!\cdots\!05}{74\!\cdots\!52}a^{15}-\frac{24\!\cdots\!49}{20\!\cdots\!56}a^{14}+\frac{15\!\cdots\!11}{34\!\cdots\!76}a^{13}-\frac{26\!\cdots\!99}{20\!\cdots\!56}a^{12}+\frac{33\!\cdots\!39}{52\!\cdots\!64}a^{11}-\frac{12\!\cdots\!55}{12\!\cdots\!68}a^{10}+\frac{49\!\cdots\!87}{17\!\cdots\!88}a^{9}-\frac{57\!\cdots\!31}{10\!\cdots\!28}a^{8}+\frac{11\!\cdots\!83}{65\!\cdots\!08}a^{7}-\frac{65\!\cdots\!41}{26\!\cdots\!32}a^{6}+\frac{91\!\cdots\!15}{10\!\cdots\!68}a^{5}-\frac{31\!\cdots\!07}{65\!\cdots\!08}a^{4}+\frac{14\!\cdots\!97}{46\!\cdots\!72}a^{3}-\frac{33\!\cdots\!05}{18\!\cdots\!88}a^{2}+\frac{39\!\cdots\!67}{46\!\cdots\!72}a-\frac{75\!\cdots\!49}{44\!\cdots\!64}$, $\frac{20\!\cdots\!49}{23\!\cdots\!84}a^{17}-\frac{10\!\cdots\!47}{43\!\cdots\!72}a^{16}+\frac{65\!\cdots\!21}{33\!\cdots\!12}a^{15}-\frac{18\!\cdots\!61}{11\!\cdots\!92}a^{14}+\frac{16\!\cdots\!63}{69\!\cdots\!52}a^{13}-\frac{62\!\cdots\!27}{17\!\cdots\!88}a^{12}+\frac{48\!\cdots\!49}{23\!\cdots\!84}a^{11}-\frac{30\!\cdots\!89}{34\!\cdots\!76}a^{10}+\frac{45\!\cdots\!59}{34\!\cdots\!76}a^{9}-\frac{72\!\cdots\!47}{57\!\cdots\!96}a^{8}+\frac{55\!\cdots\!97}{86\!\cdots\!44}a^{7}-\frac{27\!\cdots\!73}{85\!\cdots\!72}a^{6}+\frac{79\!\cdots\!61}{42\!\cdots\!36}a^{5}-\frac{25\!\cdots\!05}{72\!\cdots\!12}a^{4}+\frac{24\!\cdots\!37}{61\!\cdots\!96}a^{3}-\frac{18\!\cdots\!13}{10\!\cdots\!16}a^{2}+\frac{19\!\cdots\!05}{30\!\cdots\!48}a-\frac{57\!\cdots\!57}{22\!\cdots\!32}$, $\frac{19\!\cdots\!27}{86\!\cdots\!44}a^{17}-\frac{28\!\cdots\!89}{69\!\cdots\!52}a^{16}+\frac{28\!\cdots\!95}{49\!\cdots\!68}a^{15}-\frac{53\!\cdots\!71}{69\!\cdots\!52}a^{14}+\frac{16\!\cdots\!11}{28\!\cdots\!48}a^{13}-\frac{65\!\cdots\!61}{69\!\cdots\!52}a^{12}+\frac{19\!\cdots\!19}{34\!\cdots\!76}a^{11}-\frac{29\!\cdots\!85}{40\!\cdots\!56}a^{10}+\frac{41\!\cdots\!79}{14\!\cdots\!24}a^{9}-\frac{14\!\cdots\!33}{34\!\cdots\!76}a^{8}+\frac{56\!\cdots\!09}{43\!\cdots\!72}a^{7}-\frac{17\!\cdots\!35}{86\!\cdots\!44}a^{6}+\frac{40\!\cdots\!11}{90\!\cdots\!64}a^{5}-\frac{10\!\cdots\!79}{21\!\cdots\!36}a^{4}+\frac{75\!\cdots\!01}{30\!\cdots\!48}a^{3}-\frac{63\!\cdots\!91}{61\!\cdots\!96}a^{2}+\frac{26\!\cdots\!01}{38\!\cdots\!56}a-\frac{25\!\cdots\!43}{14\!\cdots\!88}$, $\frac{14\!\cdots\!45}{99\!\cdots\!36}a^{17}-\frac{21\!\cdots\!55}{33\!\cdots\!12}a^{16}+\frac{32\!\cdots\!05}{99\!\cdots\!36}a^{15}-\frac{41\!\cdots\!65}{58\!\cdots\!08}a^{14}+\frac{38\!\cdots\!25}{99\!\cdots\!36}a^{13}-\frac{55\!\cdots\!25}{47\!\cdots\!16}a^{12}+\frac{32\!\cdots\!55}{99\!\cdots\!36}a^{11}-\frac{55\!\cdots\!15}{99\!\cdots\!36}a^{10}+\frac{98\!\cdots\!45}{49\!\cdots\!68}a^{9}-\frac{14\!\cdots\!75}{49\!\cdots\!68}a^{8}+\frac{39\!\cdots\!55}{41\!\cdots\!64}a^{7}-\frac{31\!\cdots\!15}{17\!\cdots\!56}a^{6}+\frac{27\!\cdots\!15}{10\!\cdots\!16}a^{5}-\frac{63\!\cdots\!05}{77\!\cdots\!12}a^{4}+\frac{41\!\cdots\!15}{61\!\cdots\!96}a^{3}-\frac{24\!\cdots\!05}{88\!\cdots\!28}a^{2}+\frac{34\!\cdots\!55}{26\!\cdots\!92}a-\frac{13\!\cdots\!69}{44\!\cdots\!64}$, $\frac{14\!\cdots\!05}{14\!\cdots\!04}a^{17}-\frac{36\!\cdots\!05}{57\!\cdots\!96}a^{16}+\frac{18\!\cdots\!93}{10\!\cdots\!28}a^{15}-\frac{31\!\cdots\!73}{37\!\cdots\!76}a^{14}+\frac{19\!\cdots\!59}{10\!\cdots\!28}a^{13}-\frac{66\!\cdots\!91}{52\!\cdots\!64}a^{12}+\frac{52\!\cdots\!73}{34\!\cdots\!76}a^{11}-\frac{16\!\cdots\!61}{26\!\cdots\!32}a^{10}+\frac{24\!\cdots\!71}{30\!\cdots\!92}a^{9}-\frac{61\!\cdots\!93}{13\!\cdots\!16}a^{8}+\frac{13\!\cdots\!03}{43\!\cdots\!72}a^{7}-\frac{64\!\cdots\!07}{32\!\cdots\!04}a^{6}+\frac{29\!\cdots\!73}{32\!\cdots\!04}a^{5}-\frac{18\!\cdots\!05}{32\!\cdots\!04}a^{4}+\frac{22\!\cdots\!07}{72\!\cdots\!12}a^{3}-\frac{11\!\cdots\!05}{77\!\cdots\!12}a^{2}+\frac{24\!\cdots\!09}{46\!\cdots\!72}a-\frac{91\!\cdots\!69}{11\!\cdots\!68}$, $\frac{53\!\cdots\!89}{17\!\cdots\!24}a^{17}-\frac{62\!\cdots\!37}{20\!\cdots\!56}a^{16}+\frac{69\!\cdots\!39}{20\!\cdots\!56}a^{15}-\frac{17\!\cdots\!75}{39\!\cdots\!92}a^{14}+\frac{31\!\cdots\!37}{69\!\cdots\!52}a^{13}-\frac{94\!\cdots\!85}{20\!\cdots\!56}a^{12}+\frac{65\!\cdots\!05}{20\!\cdots\!56}a^{11}-\frac{60\!\cdots\!15}{20\!\cdots\!56}a^{10}+\frac{54\!\cdots\!45}{34\!\cdots\!76}a^{9}-\frac{17\!\cdots\!11}{10\!\cdots\!28}a^{8}+\frac{18\!\cdots\!71}{26\!\cdots\!32}a^{7}-\frac{15\!\cdots\!81}{26\!\cdots\!32}a^{6}+\frac{18\!\cdots\!43}{72\!\cdots\!12}a^{5}-\frac{51\!\cdots\!53}{32\!\cdots\!04}a^{4}+\frac{11\!\cdots\!17}{13\!\cdots\!16}a^{3}-\frac{72\!\cdots\!53}{18\!\cdots\!88}a^{2}+\frac{77\!\cdots\!69}{54\!\cdots\!32}a-\frac{78\!\cdots\!75}{30\!\cdots\!48}$ 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:  \( 23739621985693384 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 23739621985693384 \cdot 324}{2\cdot\sqrt{242459919475028479855663400956723947300369356292096}}\cr\approx \mathstrut & 3.76953387724416 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 237*x^16 - 896*x^15 + 28503*x^14 - 127218*x^13 + 2544555*x^12 - 8429568*x^11 + 155383248*x^10 - 475665960*x^9 + 7557303396*x^8 - 26162306688*x^7 + 232899383856*x^6 - 1038222554208*x^5 + 6045973402320*x^4 - 34008769056256*x^3 + 141085549266816*x^2 - 513307676719488*x + 1004539113344704)
 
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 - 6*x^17 + 237*x^16 - 896*x^15 + 28503*x^14 - 127218*x^13 + 2544555*x^12 - 8429568*x^11 + 155383248*x^10 - 475665960*x^9 + 7557303396*x^8 - 26162306688*x^7 + 232899383856*x^6 - 1038222554208*x^5 + 6045973402320*x^4 - 34008769056256*x^3 + 141085549266816*x^2 - 513307676719488*x + 1004539113344704, 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 - 6*x^17 + 237*x^16 - 896*x^15 + 28503*x^14 - 127218*x^13 + 2544555*x^12 - 8429568*x^11 + 155383248*x^10 - 475665960*x^9 + 7557303396*x^8 - 26162306688*x^7 + 232899383856*x^6 - 1038222554208*x^5 + 6045973402320*x^4 - 34008769056256*x^3 + 141085549266816*x^2 - 513307676719488*x + 1004539113344704);
 
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 - 6*x^17 + 237*x^16 - 896*x^15 + 28503*x^14 - 127218*x^13 + 2544555*x^12 - 8429568*x^11 + 155383248*x^10 - 475665960*x^9 + 7557303396*x^8 - 26162306688*x^7 + 232899383856*x^6 - 1038222554208*x^5 + 6045973402320*x^4 - 34008769056256*x^3 + 141085549266816*x^2 - 513307676719488*x + 1004539113344704);
 
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

$C_3^2:D_6$ (as 18T52):

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 solvable group of order 108
The 20 conjugacy class representatives for $C_3^2:D_6$
Character table for $C_3^2:D_6$

Intermediate fields

\(\Q(\sqrt{-34}) \), 3.1.972.2, 6.0.594140645376.27, 9.3.9023659939580352192.1

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]
 

Sibling fields

Degree 18 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.0.2669978537124037139090333900186451027656102117376.1

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 ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ R ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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 2.6.11.9$x^{6} + 4 x^{5} + 4 x^{2} + 2$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.9$x^{6} + 4 x^{5} + 4 x^{2} + 2$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.9$x^{6} + 4 x^{5} + 4 x^{2} + 2$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.6.10.3$x^{6} + 6 x^{5} + 36 x^{4} + 128 x^{3} + 297 x^{2} + 474 x + 482$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.6.10.3$x^{6} + 6 x^{5} + 36 x^{4} + 128 x^{3} + 297 x^{2} + 474 x + 482$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.6.10.3$x^{6} + 6 x^{5} + 36 x^{4} + 128 x^{3} + 297 x^{2} + 474 x + 482$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
\(7\) Copy content Toggle raw display 7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} + 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} + 7$$3$$1$$2$$C_3$$[\ ]_{3}$
\(17\) Copy content Toggle raw display 17.6.3.1$x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.12.10.2$x^{12} - 3060 x^{6} - 197676$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$