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Properties

Label 18.0.473...008.1
Degree $18$
Signature $[0, 9]$
Discriminant $-4.736\times 10^{47}$
Root discriminant \(445.28\)
Ramified primes $2,3,7,17$
Class number $3888$ (GRH)
Class group [3, 3, 6, 6, 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 + 84*x^16 - 80*x^15 + 4023*x^14 - 35010*x^13 + 336714*x^12 + 231660*x^11 + 6155667*x^10 - 9055570*x^9 + 472100244*x^8 - 1268938632*x^7 + 1311904097*x^6 - 7064712702*x^5 + 114946205706*x^4 - 1253137889748*x^3 + 6335184914100*x^2 - 19101117693096*x + 30583236418596)
 
gp: K = bnfinit(y^18 - 6*y^17 + 84*y^16 - 80*y^15 + 4023*y^14 - 35010*y^13 + 336714*y^12 + 231660*y^11 + 6155667*y^10 - 9055570*y^9 + 472100244*y^8 - 1268938632*y^7 + 1311904097*y^6 - 7064712702*y^5 + 114946205706*y^4 - 1253137889748*y^3 + 6335184914100*y^2 - 19101117693096*y + 30583236418596, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 84*x^16 - 80*x^15 + 4023*x^14 - 35010*x^13 + 336714*x^12 + 231660*x^11 + 6155667*x^10 - 9055570*x^9 + 472100244*x^8 - 1268938632*x^7 + 1311904097*x^6 - 7064712702*x^5 + 114946205706*x^4 - 1253137889748*x^3 + 6335184914100*x^2 - 19101117693096*x + 30583236418596);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 84*x^16 - 80*x^15 + 4023*x^14 - 35010*x^13 + 336714*x^12 + 231660*x^11 + 6155667*x^10 - 9055570*x^9 + 472100244*x^8 - 1268938632*x^7 + 1311904097*x^6 - 7064712702*x^5 + 114946205706*x^4 - 1253137889748*x^3 + 6335184914100*x^2 - 19101117693096*x + 30583236418596)
 

\( x^{18} - 6 x^{17} + 84 x^{16} - 80 x^{15} + 4023 x^{14} - 35010 x^{13} + 336714 x^{12} + \cdots + 30583236418596 \) 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:   \(-473554530224664999718092579993601459571033899008\) \(\medspace = -\,2^{24}\cdot 3^{30}\cdot 7^{12}\cdot 17^{13}\) 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:  \(445.28\)
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:  $2^{4/3}3^{11/6}7^{2/3}17^{5/6}\approx 732.6025438679068$
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:  \(\Q(\sqrt{-17}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{12}a^{9}-\frac{1}{4}a^{7}-\frac{1}{3}a^{3}$, $\frac{1}{48}a^{10}-\frac{1}{24}a^{9}-\frac{1}{4}a^{7}+\frac{3}{16}a^{6}-\frac{1}{8}a^{5}-\frac{11}{24}a^{4}-\frac{1}{12}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{96}a^{11}-\frac{1}{96}a^{10}+\frac{1}{48}a^{9}+\frac{3}{32}a^{7}+\frac{5}{32}a^{6}-\frac{1}{24}a^{5}-\frac{13}{48}a^{4}+\frac{1}{6}a^{3}+\frac{1}{8}a^{2}-\frac{3}{8}a-\frac{1}{8}$, $\frac{1}{45696}a^{12}+\frac{13}{7616}a^{11}+\frac{173}{45696}a^{10}-\frac{233}{7616}a^{9}+\frac{491}{15232}a^{8}-\frac{939}{7616}a^{7}-\frac{295}{6528}a^{6}-\frac{43}{448}a^{5}+\frac{3119}{22848}a^{4}-\frac{1469}{3808}a^{3}-\frac{667}{1904}a^{2}-\frac{325}{1904}a+\frac{1439}{3808}$, $\frac{1}{137088}a^{13}+\frac{1}{137088}a^{12}+\frac{355}{137088}a^{11}-\frac{305}{45696}a^{10}-\frac{755}{45696}a^{9}+\frac{2203}{45696}a^{8}-\frac{541}{19584}a^{7}+\frac{4679}{137088}a^{6}+\frac{2059}{17136}a^{5}-\frac{109}{7616}a^{4}-\frac{3413}{11424}a^{3}+\frac{5}{84}a^{2}+\frac{1455}{3808}a+\frac{5}{544}$, $\frac{1}{117347328}a^{14}-\frac{19}{58673664}a^{13}-\frac{415}{58673664}a^{12}+\frac{439}{3259648}a^{11}-\frac{154565}{19557888}a^{10}-\frac{17945}{1396992}a^{9}+\frac{2537105}{29336832}a^{8}-\frac{1407989}{14668416}a^{7}+\frac{10451465}{117347328}a^{6}-\frac{89467}{2793984}a^{5}+\frac{20605}{60928}a^{4}+\frac{4087781}{9778944}a^{3}-\frac{595453}{3259648}a^{2}-\frac{59}{13696}a+\frac{472337}{3259648}$, $\frac{1}{704083968}a^{15}-\frac{1}{234694656}a^{14}-\frac{5}{1629824}a^{13}+\frac{3649}{352041984}a^{12}-\frac{11927}{6902784}a^{11}-\frac{994093}{117347328}a^{10}-\frac{226013}{12572928}a^{9}+\frac{132065}{1150464}a^{8}+\frac{21734555}{234694656}a^{7}+\frac{10036747}{100583424}a^{6}+\frac{2959489}{19557888}a^{5}-\frac{14581813}{39115776}a^{4}+\frac{2581451}{7334208}a^{3}+\frac{325509}{6519296}a^{2}-\frac{329289}{931328}a-\frac{3156687}{6519296}$, $\frac{1}{4218871136256}a^{16}-\frac{1081}{2109435568128}a^{15}-\frac{4019}{1406290378752}a^{14}+\frac{7612987}{2109435568128}a^{13}-\frac{809395}{150673969152}a^{12}-\frac{31039805}{14648858112}a^{11}+\frac{1735701169}{301347938304}a^{10}-\frac{11292744535}{1054717784064}a^{9}-\frac{338652781}{3939188736}a^{8}-\frac{69601619617}{2109435568128}a^{7}-\frac{65183876243}{602695876608}a^{6}+\frac{35303463973}{703145189376}a^{5}+\frac{24434296325}{100449312768}a^{4}+\frac{131472005911}{351572594688}a^{3}+\frac{1440900691}{3662214528}a^{2}-\frac{6006389861}{19531810816}a+\frac{11838274543}{39063621632}$, $\frac{1}{60\!\cdots\!32}a^{17}+\frac{96\!\cdots\!65}{30\!\cdots\!16}a^{16}+\frac{19\!\cdots\!51}{31\!\cdots\!88}a^{15}-\frac{45\!\cdots\!87}{30\!\cdots\!16}a^{14}-\frac{63\!\cdots\!51}{15\!\cdots\!08}a^{13}-\frac{13\!\cdots\!99}{20\!\cdots\!64}a^{12}-\frac{48\!\cdots\!89}{30\!\cdots\!16}a^{11}-\frac{42\!\cdots\!25}{15\!\cdots\!08}a^{10}+\frac{35\!\cdots\!01}{66\!\cdots\!48}a^{9}+\frac{91\!\cdots\!53}{30\!\cdots\!16}a^{8}+\frac{24\!\cdots\!75}{35\!\cdots\!96}a^{7}-\frac{30\!\cdots\!79}{19\!\cdots\!72}a^{6}-\frac{91\!\cdots\!13}{10\!\cdots\!72}a^{5}+\frac{27\!\cdots\!05}{50\!\cdots\!36}a^{4}+\frac{12\!\cdots\!23}{41\!\cdots\!28}a^{3}-\frac{88\!\cdots\!73}{27\!\cdots\!52}a^{2}-\frac{15\!\cdots\!01}{43\!\cdots\!88}a+\frac{29\!\cdots\!59}{13\!\cdots\!76}$ 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:  $2$, $3$

Class group and class number

$C_{3}\times C_{3}\times C_{6}\times C_{6}\times C_{12}$, which has order $3888$ (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:   $\frac{66\!\cdots\!93}{22\!\cdots\!52}a^{17}-\frac{41\!\cdots\!93}{53\!\cdots\!84}a^{16}+\frac{37\!\cdots\!95}{22\!\cdots\!52}a^{15}-\frac{13\!\cdots\!23}{28\!\cdots\!44}a^{14}-\frac{34\!\cdots\!83}{23\!\cdots\!12}a^{13}-\frac{17\!\cdots\!63}{40\!\cdots\!92}a^{12}+\frac{11\!\cdots\!07}{11\!\cdots\!76}a^{11}-\frac{32\!\cdots\!83}{28\!\cdots\!44}a^{10}-\frac{22\!\cdots\!87}{22\!\cdots\!52}a^{9}-\frac{53\!\cdots\!17}{57\!\cdots\!88}a^{8}-\frac{23\!\cdots\!03}{76\!\cdots\!84}a^{7}-\frac{17\!\cdots\!13}{40\!\cdots\!92}a^{6}-\frac{16\!\cdots\!73}{11\!\cdots\!76}a^{5}-\frac{13\!\cdots\!91}{47\!\cdots\!24}a^{4}+\frac{18\!\cdots\!33}{95\!\cdots\!48}a^{3}-\frac{47\!\cdots\!93}{95\!\cdots\!48}a^{2}+\frac{89\!\cdots\!63}{19\!\cdots\!96}a-\frac{67\!\cdots\!55}{95\!\cdots\!48}$, $\frac{93\!\cdots\!01}{15\!\cdots\!96}a^{17}+\frac{67\!\cdots\!51}{35\!\cdots\!36}a^{16}+\frac{44\!\cdots\!91}{75\!\cdots\!04}a^{15}+\frac{15\!\cdots\!99}{94\!\cdots\!88}a^{14}+\frac{33\!\cdots\!39}{29\!\cdots\!84}a^{13}+\frac{29\!\cdots\!05}{26\!\cdots\!68}a^{12}+\frac{56\!\cdots\!55}{37\!\cdots\!52}a^{11}+\frac{69\!\cdots\!35}{11\!\cdots\!36}a^{10}+\frac{35\!\cdots\!93}{75\!\cdots\!04}a^{9}+\frac{18\!\cdots\!11}{53\!\cdots\!36}a^{8}+\frac{14\!\cdots\!99}{75\!\cdots\!04}a^{7}+\frac{11\!\cdots\!63}{67\!\cdots\!92}a^{6}+\frac{81\!\cdots\!97}{12\!\cdots\!84}a^{5}+\frac{58\!\cdots\!93}{22\!\cdots\!64}a^{4}+\frac{16\!\cdots\!57}{18\!\cdots\!88}a^{3}+\frac{76\!\cdots\!47}{17\!\cdots\!72}a^{2}-\frac{69\!\cdots\!81}{69\!\cdots\!88}a+\frac{17\!\cdots\!71}{34\!\cdots\!44}$, $\frac{13\!\cdots\!19}{38\!\cdots\!24}a^{17}-\frac{54\!\cdots\!29}{19\!\cdots\!52}a^{16}+\frac{54\!\cdots\!77}{18\!\cdots\!76}a^{15}+\frac{30\!\cdots\!81}{18\!\cdots\!76}a^{14}-\frac{82\!\cdots\!93}{10\!\cdots\!32}a^{13}+\frac{53\!\cdots\!97}{33\!\cdots\!96}a^{12}+\frac{15\!\cdots\!63}{94\!\cdots\!88}a^{11}+\frac{74\!\cdots\!67}{87\!\cdots\!28}a^{10}-\frac{12\!\cdots\!49}{18\!\cdots\!76}a^{9}+\frac{14\!\cdots\!91}{26\!\cdots\!68}a^{8}-\frac{46\!\cdots\!11}{20\!\cdots\!64}a^{7}+\frac{38\!\cdots\!07}{26\!\cdots\!68}a^{6}-\frac{41\!\cdots\!99}{39\!\cdots\!12}a^{5}+\frac{32\!\cdots\!91}{49\!\cdots\!92}a^{4}-\frac{47\!\cdots\!15}{15\!\cdots\!48}a^{3}+\frac{44\!\cdots\!95}{43\!\cdots\!68}a^{2}-\frac{38\!\cdots\!07}{17\!\cdots\!72}a+\frac{45\!\cdots\!59}{17\!\cdots\!72}$, $\frac{19\!\cdots\!25}{22\!\cdots\!92}a^{17}-\frac{67\!\cdots\!85}{70\!\cdots\!72}a^{16}+\frac{13\!\cdots\!87}{11\!\cdots\!08}a^{15}-\frac{20\!\cdots\!95}{41\!\cdots\!28}a^{14}+\frac{17\!\cdots\!79}{47\!\cdots\!44}a^{13}-\frac{10\!\cdots\!07}{35\!\cdots\!12}a^{12}+\frac{16\!\cdots\!91}{50\!\cdots\!36}a^{11}-\frac{39\!\cdots\!23}{53\!\cdots\!36}a^{10}+\frac{20\!\cdots\!89}{10\!\cdots\!72}a^{9}+\frac{16\!\cdots\!47}{35\!\cdots\!24}a^{8}-\frac{72\!\cdots\!43}{17\!\cdots\!48}a^{7}-\frac{18\!\cdots\!95}{17\!\cdots\!12}a^{6}+\frac{17\!\cdots\!85}{16\!\cdots\!12}a^{5}-\frac{43\!\cdots\!65}{89\!\cdots\!56}a^{4}+\frac{31\!\cdots\!13}{41\!\cdots\!28}a^{3}+\frac{99\!\cdots\!69}{13\!\cdots\!76}a^{2}-\frac{87\!\cdots\!91}{27\!\cdots\!52}a+\frac{76\!\cdots\!23}{13\!\cdots\!76}$, $\frac{56\!\cdots\!01}{14\!\cdots\!96}a^{17}-\frac{62\!\cdots\!25}{70\!\cdots\!72}a^{16}+\frac{18\!\cdots\!67}{30\!\cdots\!16}a^{15}-\frac{16\!\cdots\!35}{41\!\cdots\!28}a^{14}+\frac{73\!\cdots\!81}{94\!\cdots\!88}a^{13}-\frac{18\!\cdots\!19}{53\!\cdots\!36}a^{12}+\frac{16\!\cdots\!21}{50\!\cdots\!36}a^{11}-\frac{72\!\cdots\!01}{53\!\cdots\!36}a^{10}-\frac{17\!\cdots\!43}{30\!\cdots\!16}a^{9}-\frac{95\!\cdots\!69}{35\!\cdots\!24}a^{8}+\frac{98\!\cdots\!15}{30\!\cdots\!16}a^{7}-\frac{16\!\cdots\!37}{53\!\cdots\!36}a^{6}+\frac{58\!\cdots\!05}{50\!\cdots\!36}a^{5}+\frac{36\!\cdots\!51}{89\!\cdots\!56}a^{4}+\frac{13\!\cdots\!45}{12\!\cdots\!84}a^{3}-\frac{78\!\cdots\!83}{41\!\cdots\!28}a^{2}+\frac{21\!\cdots\!11}{27\!\cdots\!52}a-\frac{16\!\cdots\!71}{13\!\cdots\!76}$, $\frac{25\!\cdots\!21}{85\!\cdots\!76}a^{17}-\frac{20\!\cdots\!25}{85\!\cdots\!76}a^{16}+\frac{17\!\cdots\!29}{85\!\cdots\!76}a^{15}-\frac{67\!\cdots\!39}{85\!\cdots\!76}a^{14}+\frac{51\!\cdots\!57}{42\!\cdots\!88}a^{13}-\frac{36\!\cdots\!45}{21\!\cdots\!44}a^{12}+\frac{48\!\cdots\!91}{42\!\cdots\!88}a^{11}-\frac{15\!\cdots\!91}{42\!\cdots\!88}a^{10}+\frac{18\!\cdots\!49}{85\!\cdots\!76}a^{9}-\frac{28\!\cdots\!81}{85\!\cdots\!76}a^{8}+\frac{15\!\cdots\!75}{12\!\cdots\!68}a^{7}-\frac{42\!\cdots\!07}{85\!\cdots\!76}a^{6}+\frac{67\!\cdots\!13}{35\!\cdots\!24}a^{5}-\frac{16\!\cdots\!91}{14\!\cdots\!96}a^{4}+\frac{60\!\cdots\!55}{71\!\cdots\!48}a^{3}-\frac{15\!\cdots\!89}{39\!\cdots\!36}a^{2}+\frac{90\!\cdots\!13}{79\!\cdots\!72}a-\frac{17\!\cdots\!99}{11\!\cdots\!96}$, $\frac{63\!\cdots\!43}{35\!\cdots\!96}a^{17}-\frac{23\!\cdots\!35}{60\!\cdots\!32}a^{16}+\frac{26\!\cdots\!97}{20\!\cdots\!44}a^{15}+\frac{21\!\cdots\!15}{60\!\cdots\!32}a^{14}+\frac{25\!\cdots\!75}{30\!\cdots\!16}a^{13}-\frac{15\!\cdots\!53}{50\!\cdots\!36}a^{12}+\frac{14\!\cdots\!61}{30\!\cdots\!16}a^{11}+\frac{39\!\cdots\!51}{17\!\cdots\!48}a^{10}+\frac{37\!\cdots\!29}{20\!\cdots\!44}a^{9}+\frac{31\!\cdots\!73}{60\!\cdots\!32}a^{8}+\frac{61\!\cdots\!27}{60\!\cdots\!32}a^{7}+\frac{29\!\cdots\!29}{20\!\cdots\!44}a^{6}+\frac{21\!\cdots\!93}{31\!\cdots\!96}a^{5}+\frac{83\!\cdots\!19}{10\!\cdots\!72}a^{4}+\frac{36\!\cdots\!27}{16\!\cdots\!12}a^{3}-\frac{40\!\cdots\!51}{27\!\cdots\!52}a^{2}+\frac{30\!\cdots\!19}{55\!\cdots\!04}a-\frac{75\!\cdots\!55}{55\!\cdots\!04}$, $\frac{18\!\cdots\!77}{10\!\cdots\!72}a^{17}-\frac{14\!\cdots\!73}{15\!\cdots\!48}a^{16}+\frac{39\!\cdots\!37}{30\!\cdots\!16}a^{15}+\frac{27\!\cdots\!87}{27\!\cdots\!52}a^{14}+\frac{80\!\cdots\!39}{12\!\cdots\!84}a^{13}-\frac{23\!\cdots\!13}{37\!\cdots\!52}a^{12}+\frac{24\!\cdots\!23}{50\!\cdots\!36}a^{11}+\frac{78\!\cdots\!01}{62\!\cdots\!92}a^{10}+\frac{27\!\cdots\!27}{30\!\cdots\!16}a^{9}-\frac{21\!\cdots\!07}{73\!\cdots\!52}a^{8}+\frac{23\!\cdots\!49}{33\!\cdots\!24}a^{7}-\frac{13\!\cdots\!79}{75\!\cdots\!04}a^{6}-\frac{30\!\cdots\!65}{50\!\cdots\!36}a^{5}-\frac{18\!\cdots\!51}{81\!\cdots\!28}a^{4}+\frac{29\!\cdots\!89}{12\!\cdots\!84}a^{3}-\frac{73\!\cdots\!73}{41\!\cdots\!28}a^{2}+\frac{24\!\cdots\!25}{27\!\cdots\!52}a-\frac{19\!\cdots\!83}{13\!\cdots\!76}$ 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:  \( 747332298252077.8 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 747332298252077.8 \cdot 3888}{2\cdot\sqrt{473554530224664999718092579993601459571033899008}}\cr\approx \mathstrut & 32.2213572437627 \end{aligned}\] (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 + 84*x^16 - 80*x^15 + 4023*x^14 - 35010*x^13 + 336714*x^12 + 231660*x^11 + 6155667*x^10 - 9055570*x^9 + 472100244*x^8 - 1268938632*x^7 + 1311904097*x^6 - 7064712702*x^5 + 114946205706*x^4 - 1253137889748*x^3 + 6335184914100*x^2 - 19101117693096*x + 30583236418596)
 
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 + 84*x^16 - 80*x^15 + 4023*x^14 - 35010*x^13 + 336714*x^12 + 231660*x^11 + 6155667*x^10 - 9055570*x^9 + 472100244*x^8 - 1268938632*x^7 + 1311904097*x^6 - 7064712702*x^5 + 114946205706*x^4 - 1253137889748*x^3 + 6335184914100*x^2 - 19101117693096*x + 30583236418596, 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 + 84*x^16 - 80*x^15 + 4023*x^14 - 35010*x^13 + 336714*x^12 + 231660*x^11 + 6155667*x^10 - 9055570*x^9 + 472100244*x^8 - 1268938632*x^7 + 1311904097*x^6 - 7064712702*x^5 + 114946205706*x^4 - 1253137889748*x^3 + 6335184914100*x^2 - 19101117693096*x + 30583236418596);
 
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 + 84*x^16 - 80*x^15 + 4023*x^14 - 35010*x^13 + 336714*x^12 + 231660*x^11 + 6155667*x^10 - 9055570*x^9 + 472100244*x^8 - 1268938632*x^7 + 1311904097*x^6 - 7064712702*x^5 + 114946205706*x^4 - 1253137889748*x^3 + 6335184914100*x^2 - 19101117693096*x + 30583236418596);
 
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

\(\Q(\sqrt{-17}) \), 3.1.972.2, 6.0.74267580672.8, 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: 18.0.5214801830320385037285808398801662163390824448.1

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 ${\href{/padicField/5.6.0.1}{6} }^{3}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ R ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
\(7\) Copy content Toggle raw display 7.3.2.2$x^{3} + 7$$3$$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$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$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}$