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Properties

Label 18.0.167...288.1
Degree $18$
Signature $[0, 9]$
Discriminant $-1.675\times 10^{48}$
Root discriminant \(477.65\)
Ramified primes $2,3,7,19$
Class number $177147$ (GRH)
Class group [3, 3, 3, 3, 3, 3, 3, 9, 9] (GRH)
Galois group $C_9:C_6$ (as 18T18)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 - 2436*x^12 + 15903*x^11 - 106503*x^10 + 385808*x^9 - 162243*x^8 - 1489695*x^7 + 36908697*x^6 - 104067936*x^5 + 573490734*x^4 - 975956418*x^3 + 3278636388*x^2 - 2807652156*x + 5898348532)
 
gp: K = bnfinit(y^18 - 9*y^17 + 39*y^16 - 108*y^15 + 207*y^14 - 273*y^13 - 2436*y^12 + 15903*y^11 - 106503*y^10 + 385808*y^9 - 162243*y^8 - 1489695*y^7 + 36908697*y^6 - 104067936*y^5 + 573490734*y^4 - 975956418*y^3 + 3278636388*y^2 - 2807652156*y + 5898348532, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 - 2436*x^12 + 15903*x^11 - 106503*x^10 + 385808*x^9 - 162243*x^8 - 1489695*x^7 + 36908697*x^6 - 104067936*x^5 + 573490734*x^4 - 975956418*x^3 + 3278636388*x^2 - 2807652156*x + 5898348532);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 - 2436*x^12 + 15903*x^11 - 106503*x^10 + 385808*x^9 - 162243*x^8 - 1489695*x^7 + 36908697*x^6 - 104067936*x^5 + 573490734*x^4 - 975956418*x^3 + 3278636388*x^2 - 2807652156*x + 5898348532)
 

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 207 x^{14} - 273 x^{13} - 2436 x^{12} + 15903 x^{11} + \cdots + 5898348532 \) 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:   \(-1674839540438459282902886554720879596051603468288\) \(\medspace = -\,2^{12}\cdot 3^{33}\cdot 7^{16}\cdot 19^{12}\) 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:  \(477.65\)
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:  $2^{2/3}3^{109/54}7^{8/9}19^{2/3}\approx 585.4171086702478$
Ramified primes:   \(2\), \(3\), \(7\), \(19\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}$, $\frac{1}{38}a^{15}+\frac{1}{19}a^{14}+\frac{4}{19}a^{13}+\frac{8}{19}a^{11}+\frac{9}{19}a^{10}-\frac{4}{19}a^{9}-\frac{7}{19}a^{8}-\frac{8}{19}a^{7}+\frac{4}{19}a^{6}+\frac{3}{19}a^{5}-\frac{9}{19}a^{4}+\frac{17}{38}a^{3}-\frac{9}{19}a^{2}+\frac{6}{19}a+\frac{6}{19}$, $\frac{1}{30\!\cdots\!90}a^{16}-\frac{4}{15\!\cdots\!95}a^{15}-\frac{19\!\cdots\!01}{15\!\cdots\!95}a^{14}+\frac{13\!\cdots\!77}{15\!\cdots\!95}a^{13}+\frac{37\!\cdots\!36}{15\!\cdots\!95}a^{12}+\frac{49\!\cdots\!86}{15\!\cdots\!95}a^{11}+\frac{89\!\cdots\!19}{30\!\cdots\!19}a^{10}-\frac{68\!\cdots\!04}{15\!\cdots\!95}a^{9}-\frac{21\!\cdots\!69}{89\!\cdots\!35}a^{8}-\frac{40\!\cdots\!66}{89\!\cdots\!35}a^{7}-\frac{30\!\cdots\!07}{15\!\cdots\!95}a^{6}-\frac{71\!\cdots\!86}{15\!\cdots\!95}a^{5}-\frac{11\!\cdots\!93}{30\!\cdots\!90}a^{4}+\frac{59\!\cdots\!86}{15\!\cdots\!95}a^{3}-\frac{10\!\cdots\!29}{30\!\cdots\!19}a^{2}+\frac{69\!\cdots\!13}{15\!\cdots\!95}a-\frac{44\!\cdots\!48}{15\!\cdots\!95}$, $\frac{1}{34\!\cdots\!10}a^{17}+\frac{562311}{34\!\cdots\!10}a^{16}-\frac{32\!\cdots\!12}{17\!\cdots\!05}a^{15}+\frac{26\!\cdots\!41}{34\!\cdots\!10}a^{14}-\frac{42\!\cdots\!77}{34\!\cdots\!10}a^{13}+\frac{12\!\cdots\!67}{68\!\cdots\!82}a^{12}-\frac{20\!\cdots\!37}{34\!\cdots\!10}a^{11}+\frac{43\!\cdots\!47}{34\!\cdots\!10}a^{10}-\frac{82\!\cdots\!73}{34\!\cdots\!10}a^{9}+\frac{20\!\cdots\!91}{20\!\cdots\!30}a^{8}-\frac{21\!\cdots\!47}{68\!\cdots\!82}a^{7}-\frac{21\!\cdots\!93}{34\!\cdots\!10}a^{6}+\frac{34\!\cdots\!37}{17\!\cdots\!05}a^{5}+\frac{96\!\cdots\!90}{34\!\cdots\!41}a^{4}-\frac{83\!\cdots\!61}{20\!\cdots\!30}a^{3}+\frac{15\!\cdots\!42}{89\!\cdots\!95}a^{2}+\frac{14\!\cdots\!34}{17\!\cdots\!05}a-\frac{70\!\cdots\!64}{23\!\cdots\!85}$ 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

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}$, which has order $177147$ (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:   \( -\frac{8539081304383941}{3249680685137461488687535} a^{17} + \frac{145164382174526997}{6499361370274922977375070} a^{16} - \frac{241574864271440927}{3249680685137461488687535} a^{15} + \frac{7582477058748147}{68414330213420241867106} a^{14} - \frac{100394457125936127}{1299872274054984595475014} a^{13} + \frac{1144373346367940143}{6499361370274922977375070} a^{12} + \frac{2231411869837008663}{342071651067101209335530} a^{11} - \frac{249413467545902726319}{6499361370274922977375070} a^{10} + \frac{1421200407834866675749}{6499361370274922977375070} a^{9} - \frac{4495147123916233287903}{6499361370274922977375070} a^{8} - \frac{8046716405628002399301}{6499361370274922977375070} a^{7} + \frac{9470765870387327153221}{1299872274054984595475014} a^{6} - \frac{541460084975402384041131}{6499361370274922977375070} a^{5} + \frac{613027579448857705827009}{3249680685137461488687535} a^{4} - \frac{5815192154771732169935003}{6499361370274922977375070} a^{3} + \frac{3761511589663562119912014}{3249680685137461488687535} a^{2} - \frac{9573899203763918987852106}{3249680685137461488687535} a + \frac{5804676630515993811543427}{3249680685137461488687535} \)  (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:   $\frac{16\!\cdots\!53}{34\!\cdots\!10}a^{17}-\frac{68\!\cdots\!19}{17\!\cdots\!05}a^{16}+\frac{44\!\cdots\!61}{34\!\cdots\!10}a^{15}-\frac{27\!\cdots\!50}{17\!\cdots\!39}a^{14}-\frac{27\!\cdots\!09}{34\!\cdots\!41}a^{13}-\frac{41\!\cdots\!56}{17\!\cdots\!05}a^{12}-\frac{97\!\cdots\!36}{89\!\cdots\!95}a^{11}+\frac{12\!\cdots\!13}{17\!\cdots\!05}a^{10}-\frac{67\!\cdots\!03}{17\!\cdots\!05}a^{9}+\frac{10\!\cdots\!48}{10\!\cdots\!65}a^{8}+\frac{42\!\cdots\!47}{17\!\cdots\!05}a^{7}-\frac{45\!\cdots\!02}{34\!\cdots\!41}a^{6}+\frac{52\!\cdots\!09}{34\!\cdots\!10}a^{5}-\frac{53\!\cdots\!06}{17\!\cdots\!05}a^{4}+\frac{51\!\cdots\!07}{34\!\cdots\!10}a^{3}-\frac{23\!\cdots\!96}{17\!\cdots\!05}a^{2}+\frac{75\!\cdots\!24}{17\!\cdots\!05}a+\frac{47\!\cdots\!67}{17\!\cdots\!05}$, $\frac{19\!\cdots\!25}{69\!\cdots\!82}a^{16}-\frac{77\!\cdots\!00}{34\!\cdots\!41}a^{15}+\frac{47\!\cdots\!79}{69\!\cdots\!82}a^{14}-\frac{30\!\cdots\!53}{69\!\cdots\!82}a^{13}+\frac{34\!\cdots\!53}{69\!\cdots\!82}a^{12}+\frac{18\!\cdots\!71}{69\!\cdots\!82}a^{11}-\frac{16\!\cdots\!21}{69\!\cdots\!82}a^{10}-\frac{15\!\cdots\!09}{69\!\cdots\!82}a^{9}-\frac{70\!\cdots\!13}{40\!\cdots\!46}a^{8}+\frac{35\!\cdots\!89}{40\!\cdots\!46}a^{7}-\frac{14\!\cdots\!49}{69\!\cdots\!82}a^{6}+\frac{20\!\cdots\!49}{69\!\cdots\!82}a^{5}+\frac{24\!\cdots\!07}{34\!\cdots\!41}a^{4}-\frac{99\!\cdots\!71}{69\!\cdots\!82}a^{3}+\frac{43\!\cdots\!38}{34\!\cdots\!41}a^{2}-\frac{41\!\cdots\!78}{34\!\cdots\!41}a+\frac{13\!\cdots\!50}{34\!\cdots\!41}$, $\frac{21\!\cdots\!92}{17\!\cdots\!05}a^{17}-\frac{52\!\cdots\!47}{34\!\cdots\!10}a^{16}+\frac{86\!\cdots\!24}{89\!\cdots\!95}a^{15}-\frac{13\!\cdots\!99}{34\!\cdots\!10}a^{14}+\frac{43\!\cdots\!13}{34\!\cdots\!10}a^{13}-\frac{10\!\cdots\!67}{34\!\cdots\!10}a^{12}+\frac{24\!\cdots\!01}{68\!\cdots\!82}a^{11}+\frac{58\!\cdots\!03}{34\!\cdots\!10}a^{10}-\frac{76\!\cdots\!29}{34\!\cdots\!10}a^{9}+\frac{29\!\cdots\!07}{20\!\cdots\!30}a^{8}-\frac{23\!\cdots\!81}{34\!\cdots\!10}a^{7}+\frac{10\!\cdots\!47}{34\!\cdots\!10}a^{6}-\frac{30\!\cdots\!17}{34\!\cdots\!10}a^{5}+\frac{91\!\cdots\!13}{32\!\cdots\!85}a^{4}-\frac{21\!\cdots\!39}{40\!\cdots\!46}a^{3}+\frac{21\!\cdots\!87}{17\!\cdots\!05}a^{2}-\frac{20\!\cdots\!72}{17\!\cdots\!05}a+\frac{61\!\cdots\!88}{34\!\cdots\!41}$, $\frac{24\!\cdots\!25}{34\!\cdots\!41}a^{17}-\frac{70\!\cdots\!44}{17\!\cdots\!05}a^{16}+\frac{68\!\cdots\!37}{17\!\cdots\!05}a^{15}+\frac{53\!\cdots\!51}{34\!\cdots\!10}a^{14}+\frac{29\!\cdots\!14}{17\!\cdots\!05}a^{13}-\frac{92\!\cdots\!99}{10\!\cdots\!65}a^{12}-\frac{44\!\cdots\!43}{20\!\cdots\!30}a^{11}+\frac{20\!\cdots\!63}{34\!\cdots\!41}a^{10}-\frac{45\!\cdots\!23}{17\!\cdots\!05}a^{9}+\frac{99\!\cdots\!69}{20\!\cdots\!30}a^{8}+\frac{11\!\cdots\!76}{17\!\cdots\!05}a^{7}-\frac{12\!\cdots\!79}{17\!\cdots\!05}a^{6}+\frac{33\!\cdots\!13}{20\!\cdots\!30}a^{5}+\frac{38\!\cdots\!07}{17\!\cdots\!05}a^{4}+\frac{11\!\cdots\!38}{89\!\cdots\!95}a^{3}+\frac{53\!\cdots\!70}{34\!\cdots\!41}a^{2}+\frac{34\!\cdots\!36}{17\!\cdots\!05}a+\frac{12\!\cdots\!79}{17\!\cdots\!05}$, $\frac{24\!\cdots\!25}{34\!\cdots\!41}a^{17}-\frac{13\!\cdots\!81}{17\!\cdots\!05}a^{16}+\frac{58\!\cdots\!33}{17\!\cdots\!05}a^{15}-\frac{20\!\cdots\!01}{34\!\cdots\!10}a^{14}+\frac{17\!\cdots\!16}{17\!\cdots\!05}a^{13}-\frac{10\!\cdots\!49}{34\!\cdots\!10}a^{12}-\frac{45\!\cdots\!49}{34\!\cdots\!10}a^{11}+\frac{53\!\cdots\!09}{34\!\cdots\!41}a^{10}-\frac{29\!\cdots\!99}{34\!\cdots\!10}a^{9}+\frac{58\!\cdots\!21}{20\!\cdots\!30}a^{8}+\frac{14\!\cdots\!64}{17\!\cdots\!05}a^{7}-\frac{11\!\cdots\!47}{34\!\cdots\!10}a^{6}+\frac{90\!\cdots\!99}{34\!\cdots\!10}a^{5}-\frac{17\!\cdots\!52}{17\!\cdots\!05}a^{4}+\frac{62\!\cdots\!13}{20\!\cdots\!30}a^{3}-\frac{24\!\cdots\!26}{34\!\cdots\!41}a^{2}+\frac{16\!\cdots\!74}{17\!\cdots\!05}a-\frac{21\!\cdots\!89}{17\!\cdots\!05}$, $\frac{19\!\cdots\!29}{34\!\cdots\!10}a^{17}-\frac{85\!\cdots\!52}{17\!\cdots\!05}a^{16}+\frac{78\!\cdots\!83}{34\!\cdots\!10}a^{15}-\frac{23\!\cdots\!29}{68\!\cdots\!82}a^{14}+\frac{24\!\cdots\!35}{34\!\cdots\!41}a^{13}+\frac{29\!\cdots\!29}{34\!\cdots\!10}a^{12}-\frac{20\!\cdots\!59}{34\!\cdots\!10}a^{11}+\frac{19\!\cdots\!19}{17\!\cdots\!05}a^{10}-\frac{20\!\cdots\!23}{34\!\cdots\!10}a^{9}+\frac{25\!\cdots\!43}{20\!\cdots\!30}a^{8}-\frac{57\!\cdots\!69}{17\!\cdots\!05}a^{7}-\frac{14\!\cdots\!77}{40\!\cdots\!46}a^{6}+\frac{20\!\cdots\!86}{17\!\cdots\!05}a^{5}-\frac{14\!\cdots\!43}{17\!\cdots\!05}a^{4}+\frac{28\!\cdots\!93}{17\!\cdots\!05}a^{3}-\frac{10\!\cdots\!93}{17\!\cdots\!05}a^{2}+\frac{59\!\cdots\!86}{10\!\cdots\!65}a-\frac{23\!\cdots\!69}{17\!\cdots\!05}$, $\frac{11\!\cdots\!05}{35\!\cdots\!78}a^{17}-\frac{37\!\cdots\!18}{20\!\cdots\!73}a^{16}+\frac{23\!\cdots\!17}{68\!\cdots\!82}a^{15}-\frac{12\!\cdots\!05}{68\!\cdots\!82}a^{14}+\frac{52\!\cdots\!99}{34\!\cdots\!41}a^{13}+\frac{17\!\cdots\!13}{34\!\cdots\!41}a^{12}-\frac{72\!\cdots\!69}{68\!\cdots\!82}a^{11}+\frac{21\!\cdots\!78}{34\!\cdots\!41}a^{10}-\frac{26\!\cdots\!55}{34\!\cdots\!41}a^{9}+\frac{25\!\cdots\!99}{40\!\cdots\!46}a^{8}+\frac{30\!\cdots\!21}{34\!\cdots\!41}a^{7}-\frac{36\!\cdots\!93}{34\!\cdots\!41}a^{6}+\frac{14\!\cdots\!12}{34\!\cdots\!41}a^{5}-\frac{16\!\cdots\!65}{64\!\cdots\!97}a^{4}+\frac{30\!\cdots\!11}{68\!\cdots\!82}a^{3}-\frac{33\!\cdots\!33}{20\!\cdots\!73}a^{2}+\frac{47\!\cdots\!96}{34\!\cdots\!41}a-\frac{11\!\cdots\!66}{34\!\cdots\!41}$, $\frac{13\!\cdots\!09}{34\!\cdots\!10}a^{17}-\frac{23\!\cdots\!29}{34\!\cdots\!10}a^{16}+\frac{13\!\cdots\!63}{34\!\cdots\!10}a^{15}-\frac{39\!\cdots\!74}{34\!\cdots\!41}a^{14}+\frac{12\!\cdots\!21}{68\!\cdots\!82}a^{13}-\frac{27\!\cdots\!63}{17\!\cdots\!05}a^{12}-\frac{16\!\cdots\!67}{17\!\cdots\!05}a^{11}+\frac{51\!\cdots\!13}{34\!\cdots\!10}a^{10}-\frac{72\!\cdots\!86}{89\!\cdots\!95}a^{9}+\frac{39\!\cdots\!44}{10\!\cdots\!65}a^{8}-\frac{24\!\cdots\!73}{34\!\cdots\!10}a^{7}-\frac{85\!\cdots\!71}{34\!\cdots\!41}a^{6}+\frac{71\!\cdots\!47}{34\!\cdots\!10}a^{5}-\frac{24\!\cdots\!38}{17\!\cdots\!05}a^{4}+\frac{12\!\cdots\!31}{34\!\cdots\!10}a^{3}-\frac{24\!\cdots\!23}{17\!\cdots\!05}a^{2}+\frac{29\!\cdots\!42}{17\!\cdots\!05}a-\frac{15\!\cdots\!98}{32\!\cdots\!85}$ 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:  \( 3161751653565.267 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 3161751653565.267 \cdot 177147}{6\cdot\sqrt{1674839540438459282902886554720879596051603468288}}\cr\approx \mathstrut & 1.10088642993148 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 - 2436*x^12 + 15903*x^11 - 106503*x^10 + 385808*x^9 - 162243*x^8 - 1489695*x^7 + 36908697*x^6 - 104067936*x^5 + 573490734*x^4 - 975956418*x^3 + 3278636388*x^2 - 2807652156*x + 5898348532)
 
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 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 - 2436*x^12 + 15903*x^11 - 106503*x^10 + 385808*x^9 - 162243*x^8 - 1489695*x^7 + 36908697*x^6 - 104067936*x^5 + 573490734*x^4 - 975956418*x^3 + 3278636388*x^2 - 2807652156*x + 5898348532, 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 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 - 2436*x^12 + 15903*x^11 - 106503*x^10 + 385808*x^9 - 162243*x^8 - 1489695*x^7 + 36908697*x^6 - 104067936*x^5 + 573490734*x^4 - 975956418*x^3 + 3278636388*x^2 - 2807652156*x + 5898348532);
 
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 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 - 2436*x^12 + 15903*x^11 - 106503*x^10 + 385808*x^9 - 162243*x^8 - 1489695*x^7 + 36908697*x^6 - 104067936*x^5 + 573490734*x^4 - 975956418*x^3 + 3278636388*x^2 - 2807652156*x + 5898348532);
 
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

$C_9:C_6$ (as 18T18):

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 54
The 10 conjugacy class representatives for $C_9:C_6$
Character table for $C_9:C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1323.1 x3, 6.0.5250987.1, 9.1.747181267707388232384064.1 x3

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 9 sibling: data not computed
Degree 27 sibling: data not computed
Minimal sibling: 9.1.747181267707388232384064.1

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 ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ R ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.9.0.1}{9} }^{2}$ ${\href{/padicField/37.9.0.1}{9} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(3\) Copy content Toggle raw display Deg $18$$18$$1$$33$
\(7\) Copy content Toggle raw display 7.9.8.2$x^{9} + 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
7.9.8.2$x^{9} + 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{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}$