Properties

Label 18.0.376...192.1
Degree $18$
Signature $[0, 9]$
Discriminant $-3.762\times 10^{28}$
Root discriminant \(38.68\)
Ramified primes $2,3,67$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^16 - 16*x^15 - 20*x^14 + 152*x^13 + 370*x^12 + 32*x^11 - 2786*x^10 - 3960*x^9 + 13300*x^8 + 14472*x^7 - 12096*x^6 - 79632*x^5 + 22194*x^4 + 192240*x^3 - 108675*x^2 - 139320*x + 106677)
 
gp: K = bnfinit(y^18 - 5*y^16 - 16*y^15 - 20*y^14 + 152*y^13 + 370*y^12 + 32*y^11 - 2786*y^10 - 3960*y^9 + 13300*y^8 + 14472*y^7 - 12096*y^6 - 79632*y^5 + 22194*y^4 + 192240*y^3 - 108675*y^2 - 139320*y + 106677, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 5*x^16 - 16*x^15 - 20*x^14 + 152*x^13 + 370*x^12 + 32*x^11 - 2786*x^10 - 3960*x^9 + 13300*x^8 + 14472*x^7 - 12096*x^6 - 79632*x^5 + 22194*x^4 + 192240*x^3 - 108675*x^2 - 139320*x + 106677);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 5*x^16 - 16*x^15 - 20*x^14 + 152*x^13 + 370*x^12 + 32*x^11 - 2786*x^10 - 3960*x^9 + 13300*x^8 + 14472*x^7 - 12096*x^6 - 79632*x^5 + 22194*x^4 + 192240*x^3 - 108675*x^2 - 139320*x + 106677)
 

\( x^{18} - 5 x^{16} - 16 x^{15} - 20 x^{14} + 152 x^{13} + 370 x^{12} + 32 x^{11} - 2786 x^{10} - 3960 x^{9} + 13300 x^{8} + 14472 x^{7} - 12096 x^{6} - 79632 x^{5} + \cdots + 106677 \) 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:   \(-37621770755235979566165000192\) \(\medspace = -\,2^{20}\cdot 3^{9}\cdot 67^{10}\) 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:  \(38.68\)
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:  $2^{4/3}3^{1/2}67^{5/6}\approx 145.0985114774686$
Ramified primes:   \(2\), \(3\), \(67\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{4}+\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}+\frac{1}{6}a^{3}-\frac{1}{2}a$, $\frac{1}{18}a^{8}+\frac{1}{18}a^{7}+\frac{1}{18}a^{6}+\frac{7}{18}a^{5}-\frac{5}{18}a^{4}+\frac{1}{18}a^{3}-\frac{1}{6}a^{2}+\frac{1}{6}a$, $\frac{1}{18}a^{9}+\frac{1}{3}a^{5}-\frac{2}{9}a^{3}-\frac{1}{6}a$, $\frac{1}{18}a^{10}+\frac{4}{9}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{18}a^{11}+\frac{4}{9}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{108}a^{12}-\frac{1}{54}a^{10}-\frac{1}{54}a^{9}+\frac{1}{108}a^{8}-\frac{1}{54}a^{7}-\frac{1}{18}a^{6}+\frac{23}{54}a^{5}+\frac{5}{12}a^{4}+\frac{7}{18}a^{3}+\frac{1}{3}a-\frac{1}{4}$, $\frac{1}{324}a^{13}+\frac{1}{81}a^{11}-\frac{1}{162}a^{10}+\frac{1}{324}a^{9}-\frac{1}{162}a^{8}+\frac{1}{27}a^{7}-\frac{2}{81}a^{6}+\frac{37}{108}a^{5}-\frac{1}{27}a^{4}+\frac{2}{9}a^{3}+\frac{5}{18}a^{2}+\frac{1}{12}a-\frac{1}{2}$, $\frac{1}{324}a^{14}+\frac{1}{324}a^{12}-\frac{1}{162}a^{11}+\frac{7}{324}a^{10}+\frac{1}{81}a^{9}-\frac{1}{36}a^{8}-\frac{5}{81}a^{7}+\frac{1}{108}a^{6}+\frac{4}{27}a^{5}-\frac{1}{4}a^{4}-\frac{1}{6}a^{3}-\frac{1}{12}a^{2}+\frac{1}{4}$, $\frac{1}{648}a^{15}-\frac{1}{648}a^{14}-\frac{1}{648}a^{13}-\frac{1}{216}a^{12}+\frac{1}{648}a^{11}-\frac{17}{648}a^{10}-\frac{5}{216}a^{9}+\frac{11}{648}a^{8}-\frac{37}{648}a^{7}+\frac{25}{648}a^{6}+\frac{5}{72}a^{5}-\frac{79}{216}a^{4}-\frac{17}{72}a^{3}-\frac{11}{72}a^{2}-\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{1944}a^{16}-\frac{1}{1944}a^{15}-\frac{1}{1944}a^{14}-\frac{1}{648}a^{13}+\frac{1}{1944}a^{12}-\frac{53}{1944}a^{11}+\frac{7}{648}a^{10}+\frac{47}{1944}a^{9}-\frac{1}{1944}a^{8}-\frac{155}{1944}a^{7}-\frac{5}{72}a^{6}+\frac{125}{648}a^{5}-\frac{101}{216}a^{4}+\frac{61}{216}a^{3}+\frac{7}{24}a^{2}+\frac{3}{8}a$, $\frac{1}{27\!\cdots\!72}a^{17}+\frac{32\!\cdots\!93}{30\!\cdots\!08}a^{16}-\frac{88\!\cdots\!89}{27\!\cdots\!72}a^{15}+\frac{33\!\cdots\!39}{27\!\cdots\!72}a^{14}+\frac{22\!\cdots\!93}{27\!\cdots\!72}a^{13}-\frac{43\!\cdots\!69}{27\!\cdots\!72}a^{12}+\frac{92\!\cdots\!29}{27\!\cdots\!72}a^{11}+\frac{91\!\cdots\!73}{27\!\cdots\!72}a^{10}+\frac{54\!\cdots\!95}{27\!\cdots\!72}a^{9}+\frac{11\!\cdots\!07}{92\!\cdots\!24}a^{8}-\frac{19\!\cdots\!67}{27\!\cdots\!72}a^{7}+\frac{39\!\cdots\!39}{92\!\cdots\!24}a^{6}-\frac{32\!\cdots\!67}{92\!\cdots\!24}a^{5}+\frac{10\!\cdots\!21}{30\!\cdots\!08}a^{4}-\frac{14\!\cdots\!17}{30\!\cdots\!08}a^{3}-\frac{60\!\cdots\!37}{34\!\cdots\!12}a^{2}-\frac{45\!\cdots\!93}{28\!\cdots\!26}a+\frac{23\!\cdots\!69}{56\!\cdots\!52}$ 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

$C_{3}$, which has order $3$ (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:   \( \frac{13149390871}{101161545473124} a^{17} + \frac{17105954305}{202323090946248} a^{16} - \frac{108972610469}{202323090946248} a^{15} - \frac{170418584869}{67441030315416} a^{14} - \frac{925569130669}{202323090946248} a^{13} + \frac{3251323147535}{202323090946248} a^{12} + \frac{3976104782173}{67441030315416} a^{11} + \frac{11085754317583}{202323090946248} a^{10} - \frac{63407050411535}{202323090946248} a^{9} - \frac{151288649895223}{202323090946248} a^{8} + \frac{69728492846111}{67441030315416} a^{7} + \frac{166325193411949}{67441030315416} a^{6} + \frac{25138580073269}{22480343438472} a^{5} - \frac{208906423498651}{22480343438472} a^{4} - \frac{3450147519897}{832605312536} a^{3} + \frac{41748836071829}{2497815937608} a^{2} + \frac{3757648791533}{2497815937608} a - \frac{3608743361031}{416302656268} \)  (order $6$) 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:   $\frac{68\!\cdots\!48}{75\!\cdots\!87}a^{17}+\frac{28\!\cdots\!27}{30\!\cdots\!48}a^{16}-\frac{55\!\cdots\!47}{15\!\cdots\!74}a^{15}-\frac{62\!\cdots\!79}{33\!\cdots\!72}a^{14}-\frac{28\!\cdots\!89}{75\!\cdots\!87}a^{13}+\frac{30\!\cdots\!51}{30\!\cdots\!48}a^{12}+\frac{22\!\cdots\!29}{50\!\cdots\!58}a^{11}+\frac{15\!\cdots\!73}{30\!\cdots\!48}a^{10}-\frac{31\!\cdots\!57}{15\!\cdots\!74}a^{9}-\frac{17\!\cdots\!07}{30\!\cdots\!48}a^{8}+\frac{10\!\cdots\!17}{16\!\cdots\!86}a^{7}+\frac{20\!\cdots\!19}{10\!\cdots\!16}a^{6}+\frac{87\!\cdots\!93}{83\!\cdots\!43}a^{5}-\frac{21\!\cdots\!73}{33\!\cdots\!72}a^{4}-\frac{30\!\cdots\!25}{62\!\cdots\!18}a^{3}+\frac{14\!\cdots\!03}{11\!\cdots\!24}a^{2}+\frac{23\!\cdots\!93}{62\!\cdots\!18}a-\frac{54\!\cdots\!13}{62\!\cdots\!18}$, $\frac{72\!\cdots\!97}{11\!\cdots\!53}a^{17}+\frac{54\!\cdots\!73}{92\!\cdots\!24}a^{16}-\frac{22\!\cdots\!75}{92\!\cdots\!24}a^{15}-\frac{11\!\cdots\!35}{92\!\cdots\!24}a^{14}-\frac{23\!\cdots\!15}{92\!\cdots\!24}a^{13}+\frac{64\!\cdots\!31}{92\!\cdots\!24}a^{12}+\frac{27\!\cdots\!61}{92\!\cdots\!24}a^{11}+\frac{10\!\cdots\!77}{30\!\cdots\!08}a^{10}-\frac{43\!\cdots\!15}{30\!\cdots\!08}a^{9}-\frac{35\!\cdots\!19}{92\!\cdots\!24}a^{8}+\frac{39\!\cdots\!07}{92\!\cdots\!24}a^{7}+\frac{43\!\cdots\!21}{34\!\cdots\!12}a^{6}+\frac{72\!\cdots\!11}{10\!\cdots\!36}a^{5}-\frac{43\!\cdots\!39}{10\!\cdots\!36}a^{4}-\frac{93\!\cdots\!59}{34\!\cdots\!12}a^{3}+\frac{27\!\cdots\!99}{34\!\cdots\!12}a^{2}+\frac{68\!\cdots\!55}{34\!\cdots\!12}a-\frac{28\!\cdots\!09}{56\!\cdots\!52}$, $\frac{57\!\cdots\!07}{27\!\cdots\!72}a^{17}+\frac{53\!\cdots\!05}{17\!\cdots\!56}a^{16}-\frac{36\!\cdots\!25}{27\!\cdots\!72}a^{15}-\frac{11\!\cdots\!35}{27\!\cdots\!72}a^{14}-\frac{13\!\cdots\!85}{27\!\cdots\!72}a^{13}+\frac{10\!\cdots\!15}{27\!\cdots\!72}a^{12}+\frac{29\!\cdots\!89}{27\!\cdots\!72}a^{11}+\frac{60\!\cdots\!97}{27\!\cdots\!72}a^{10}-\frac{20\!\cdots\!95}{27\!\cdots\!72}a^{9}-\frac{12\!\cdots\!73}{92\!\cdots\!24}a^{8}+\frac{74\!\cdots\!49}{27\!\cdots\!72}a^{7}+\frac{51\!\cdots\!37}{92\!\cdots\!24}a^{6}-\frac{37\!\cdots\!57}{92\!\cdots\!24}a^{5}-\frac{66\!\cdots\!91}{30\!\cdots\!08}a^{4}-\frac{28\!\cdots\!45}{30\!\cdots\!08}a^{3}+\frac{15\!\cdots\!43}{34\!\cdots\!12}a^{2}+\frac{28\!\cdots\!01}{42\!\cdots\!39}a-\frac{36\!\cdots\!49}{11\!\cdots\!04}$, $\frac{49\!\cdots\!23}{13\!\cdots\!36}a^{17}+\frac{72\!\cdots\!91}{69\!\cdots\!18}a^{16}-\frac{32\!\cdots\!95}{27\!\cdots\!72}a^{15}-\frac{17\!\cdots\!33}{34\!\cdots\!12}a^{14}-\frac{30\!\cdots\!77}{27\!\cdots\!72}a^{13}+\frac{10\!\cdots\!59}{27\!\cdots\!72}a^{12}+\frac{11\!\cdots\!35}{92\!\cdots\!24}a^{11}+\frac{30\!\cdots\!99}{27\!\cdots\!72}a^{10}-\frac{18\!\cdots\!91}{27\!\cdots\!72}a^{9}-\frac{33\!\cdots\!23}{27\!\cdots\!72}a^{8}+\frac{25\!\cdots\!29}{92\!\cdots\!24}a^{7}+\frac{19\!\cdots\!19}{92\!\cdots\!24}a^{6}+\frac{27\!\cdots\!09}{30\!\cdots\!08}a^{5}-\frac{38\!\cdots\!19}{30\!\cdots\!08}a^{4}+\frac{23\!\cdots\!03}{34\!\cdots\!12}a^{3}+\frac{77\!\cdots\!87}{10\!\cdots\!36}a^{2}-\frac{39\!\cdots\!51}{34\!\cdots\!12}a+\frac{16\!\cdots\!29}{11\!\cdots\!04}$, $\frac{59\!\cdots\!89}{27\!\cdots\!72}a^{17}+\frac{36\!\cdots\!67}{13\!\cdots\!36}a^{16}-\frac{21\!\cdots\!11}{27\!\cdots\!72}a^{15}-\frac{13\!\cdots\!71}{30\!\cdots\!08}a^{14}-\frac{27\!\cdots\!51}{27\!\cdots\!72}a^{13}+\frac{59\!\cdots\!99}{27\!\cdots\!72}a^{12}+\frac{36\!\cdots\!63}{34\!\cdots\!12}a^{11}+\frac{38\!\cdots\!05}{27\!\cdots\!72}a^{10}-\frac{12\!\cdots\!49}{27\!\cdots\!72}a^{9}-\frac{39\!\cdots\!35}{27\!\cdots\!72}a^{8}+\frac{10\!\cdots\!33}{92\!\cdots\!24}a^{7}+\frac{42\!\cdots\!09}{92\!\cdots\!24}a^{6}+\frac{94\!\cdots\!91}{30\!\cdots\!08}a^{5}-\frac{42\!\cdots\!59}{30\!\cdots\!08}a^{4}-\frac{12\!\cdots\!53}{10\!\cdots\!36}a^{3}+\frac{27\!\cdots\!13}{10\!\cdots\!36}a^{2}+\frac{84\!\cdots\!51}{85\!\cdots\!78}a-\frac{21\!\cdots\!25}{11\!\cdots\!04}$, $\frac{24\!\cdots\!51}{69\!\cdots\!18}a^{17}+\frac{88\!\cdots\!31}{13\!\cdots\!36}a^{16}-\frac{20\!\cdots\!09}{27\!\cdots\!72}a^{15}-\frac{22\!\cdots\!39}{30\!\cdots\!08}a^{14}-\frac{57\!\cdots\!35}{27\!\cdots\!72}a^{13}+\frac{50\!\cdots\!25}{27\!\cdots\!72}a^{12}+\frac{16\!\cdots\!09}{92\!\cdots\!24}a^{11}+\frac{93\!\cdots\!01}{27\!\cdots\!72}a^{10}-\frac{12\!\cdots\!21}{27\!\cdots\!72}a^{9}-\frac{67\!\cdots\!53}{27\!\cdots\!72}a^{8}+\frac{13\!\cdots\!31}{92\!\cdots\!24}a^{7}+\frac{60\!\cdots\!09}{92\!\cdots\!24}a^{6}+\frac{27\!\cdots\!03}{30\!\cdots\!08}a^{5}-\frac{43\!\cdots\!33}{30\!\cdots\!08}a^{4}-\frac{82\!\cdots\!87}{34\!\cdots\!12}a^{3}+\frac{80\!\cdots\!63}{34\!\cdots\!12}a^{2}+\frac{59\!\cdots\!73}{34\!\cdots\!12}a-\frac{17\!\cdots\!43}{11\!\cdots\!04}$, $\frac{15\!\cdots\!35}{69\!\cdots\!18}a^{17}+\frac{39\!\cdots\!59}{13\!\cdots\!36}a^{16}-\frac{78\!\cdots\!15}{92\!\cdots\!24}a^{15}-\frac{12\!\cdots\!83}{27\!\cdots\!72}a^{14}-\frac{27\!\cdots\!29}{27\!\cdots\!72}a^{13}+\frac{20\!\cdots\!43}{92\!\cdots\!24}a^{12}+\frac{30\!\cdots\!15}{27\!\cdots\!72}a^{11}+\frac{38\!\cdots\!89}{27\!\cdots\!72}a^{10}-\frac{42\!\cdots\!21}{92\!\cdots\!24}a^{9}-\frac{40\!\cdots\!09}{27\!\cdots\!72}a^{8}+\frac{35\!\cdots\!57}{27\!\cdots\!72}a^{7}+\frac{45\!\cdots\!89}{92\!\cdots\!24}a^{6}+\frac{27\!\cdots\!75}{92\!\cdots\!24}a^{5}-\frac{16\!\cdots\!75}{11\!\cdots\!04}a^{4}-\frac{41\!\cdots\!95}{30\!\cdots\!08}a^{3}+\frac{34\!\cdots\!69}{11\!\cdots\!04}a^{2}+\frac{13\!\cdots\!09}{11\!\cdots\!04}a-\frac{24\!\cdots\!11}{11\!\cdots\!04}$, $\frac{53\!\cdots\!18}{11\!\cdots\!53}a^{17}+\frac{27\!\cdots\!73}{27\!\cdots\!72}a^{16}-\frac{26\!\cdots\!91}{27\!\cdots\!72}a^{15}-\frac{28\!\cdots\!49}{27\!\cdots\!72}a^{14}-\frac{27\!\cdots\!81}{92\!\cdots\!24}a^{13}+\frac{56\!\cdots\!43}{27\!\cdots\!72}a^{12}+\frac{69\!\cdots\!05}{27\!\cdots\!72}a^{11}+\frac{46\!\cdots\!43}{92\!\cdots\!24}a^{10}-\frac{14\!\cdots\!49}{27\!\cdots\!72}a^{9}-\frac{97\!\cdots\!03}{27\!\cdots\!72}a^{8}-\frac{12\!\cdots\!09}{27\!\cdots\!72}a^{7}+\frac{10\!\cdots\!59}{10\!\cdots\!36}a^{6}+\frac{12\!\cdots\!33}{92\!\cdots\!24}a^{5}-\frac{52\!\cdots\!03}{30\!\cdots\!08}a^{4}-\frac{12\!\cdots\!45}{30\!\cdots\!08}a^{3}+\frac{10\!\cdots\!03}{34\!\cdots\!12}a^{2}+\frac{40\!\cdots\!75}{11\!\cdots\!04}a-\frac{15\!\cdots\!41}{56\!\cdots\!52}$ 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:  \( 306761663.9033038 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 306761663.9033038 \cdot 3}{6\cdot\sqrt{37621770755235979566165000192}}\cr\approx \mathstrut & 12.0689722279917 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^16 - 16*x^15 - 20*x^14 + 152*x^13 + 370*x^12 + 32*x^11 - 2786*x^10 - 3960*x^9 + 13300*x^8 + 14472*x^7 - 12096*x^6 - 79632*x^5 + 22194*x^4 + 192240*x^3 - 108675*x^2 - 139320*x + 106677)
 
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 - 5*x^16 - 16*x^15 - 20*x^14 + 152*x^13 + 370*x^12 + 32*x^11 - 2786*x^10 - 3960*x^9 + 13300*x^8 + 14472*x^7 - 12096*x^6 - 79632*x^5 + 22194*x^4 + 192240*x^3 - 108675*x^2 - 139320*x + 106677, 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 - 5*x^16 - 16*x^15 - 20*x^14 + 152*x^13 + 370*x^12 + 32*x^11 - 2786*x^10 - 3960*x^9 + 13300*x^8 + 14472*x^7 - 12096*x^6 - 79632*x^5 + 22194*x^4 + 192240*x^3 - 108675*x^2 - 139320*x + 106677);
 
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 - 5*x^16 - 16*x^15 - 20*x^14 + 152*x^13 + 370*x^12 + 32*x^11 - 2786*x^10 - 3960*x^9 + 13300*x^8 + 14472*x^7 - 12096*x^6 - 79632*x^5 + 22194*x^4 + 192240*x^3 - 108675*x^2 - 139320*x + 106677);
 
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

$C_3\times S_3^2$ (as 18T46):

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 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.804.1, 6.0.1939248.2, 6.0.1939248.3

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 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.0.19400185409620625915904.1

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 ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.16.3$x^{12} + 4 x^{11} + 8 x^{10} + 10 x^{9} + 8 x^{8} + 4 x^{7} + 2 x^{6} - 4 x^{5} - 4 x^{4} + 4 x^{3} + 4$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
\(67\) Copy content Toggle raw display 67.3.2.2$x^{3} + 268$$3$$1$$2$$C_3$$[\ ]_{3}$
67.3.0.1$x^{3} + 6 x + 65$$1$$3$$0$$C_3$$[\ ]^{3}$
67.6.5.3$x^{6} + 469$$6$$1$$5$$C_6$$[\ ]_{6}$
67.6.3.1$x^{6} + 26934 x^{2} - 19549595$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$