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Properties

Label 18.0.249...368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-2.496\times 10^{28}$
Root discriminant \(37.81\)
Ramified primes $2,3,17$
Class number $9$ (GRH)
Class group [3, 3] (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 + 24*x^14 - 132*x^13 + 13*x^12 - 120*x^11 - 348*x^10 + 864*x^9 + 2532*x^8 + 1536*x^7 + 4219*x^6 - 2976*x^5 - 5622*x^4 + 7200*x^3 + 16140*x^2 + 10440*x + 2523)
 
gp: K = bnfinit(y^18 + 6*y^16 + 24*y^14 - 132*y^13 + 13*y^12 - 120*y^11 - 348*y^10 + 864*y^9 + 2532*y^8 + 1536*y^7 + 4219*y^6 - 2976*y^5 - 5622*y^4 + 7200*y^3 + 16140*y^2 + 10440*y + 2523, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 6*x^16 + 24*x^14 - 132*x^13 + 13*x^12 - 120*x^11 - 348*x^10 + 864*x^9 + 2532*x^8 + 1536*x^7 + 4219*x^6 - 2976*x^5 - 5622*x^4 + 7200*x^3 + 16140*x^2 + 10440*x + 2523);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 6*x^16 + 24*x^14 - 132*x^13 + 13*x^12 - 120*x^11 - 348*x^10 + 864*x^9 + 2532*x^8 + 1536*x^7 + 4219*x^6 - 2976*x^5 - 5622*x^4 + 7200*x^3 + 16140*x^2 + 10440*x + 2523)
 

\( x^{18} + 6 x^{16} + 24 x^{14} - 132 x^{13} + 13 x^{12} - 120 x^{11} - 348 x^{10} + 864 x^{9} + \cdots + 2523 \) 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:   \(-24962803242374579500132282368\) \(\medspace = -\,2^{12}\cdot 3^{21}\cdot 17^{12}\) 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:  \(37.81\)
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:  $2^{2/3}3^{7/6}17^{2/3}\approx 37.81180491429508$
Ramified primes:   \(2\), \(3\), \(17\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{10}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}+\frac{3}{10}a^{8}+\frac{1}{5}a^{7}-\frac{3}{10}a^{5}+\frac{2}{5}a^{4}-\frac{1}{10}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{10}a^{12}-\frac{1}{10}a^{10}-\frac{1}{10}a^{9}-\frac{1}{5}a^{8}-\frac{1}{10}a^{7}-\frac{3}{10}a^{6}-\frac{1}{5}a^{5}+\frac{3}{10}a^{4}-\frac{1}{10}a^{3}+\frac{2}{5}a^{2}-\frac{1}{10}a+\frac{2}{5}$, $\frac{1}{10}a^{13}+\frac{1}{5}a^{10}+\frac{1}{10}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{3}{10}a^{6}-\frac{1}{5}a^{4}-\frac{1}{10}a^{3}-\frac{1}{5}a^{2}-\frac{1}{2}a-\frac{3}{10}$, $\frac{1}{10}a^{14}+\frac{1}{10}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{2}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{3}{10}a^{3}-\frac{3}{10}a^{2}+\frac{1}{10}$, $\frac{1}{340}a^{15}-\frac{3}{85}a^{14}-\frac{3}{68}a^{12}-\frac{7}{170}a^{11}-\frac{13}{170}a^{10}-\frac{33}{170}a^{9}-\frac{37}{170}a^{8}+\frac{1}{10}a^{7}+\frac{1}{85}a^{6}-\frac{9}{34}a^{5}+\frac{1}{34}a^{4}-\frac{99}{340}a^{3}-\frac{1}{34}a^{2}-\frac{49}{170}a-\frac{25}{68}$, $\frac{1}{4745227928540}a^{16}-\frac{3682763981}{4745227928540}a^{15}-\frac{8467598443}{1186306982135}a^{14}-\frac{141468650659}{4745227928540}a^{13}-\frac{155338897861}{4745227928540}a^{12}+\frac{3903031856}{237261396427}a^{11}+\frac{329052305821}{2372613964270}a^{10}-\frac{225527895291}{2372613964270}a^{9}-\frac{10653081986}{1186306982135}a^{8}-\frac{155947116599}{1186306982135}a^{7}-\frac{50495044843}{1186306982135}a^{6}+\frac{190918258136}{1186306982135}a^{5}+\frac{2066371308373}{4745227928540}a^{4}-\frac{1603825484733}{4745227928540}a^{3}+\frac{493910494333}{1186306982135}a^{2}-\frac{217317333281}{4745227928540}a-\frac{18180508019}{163628549260}$, $\frac{1}{74\!\cdots\!00}a^{17}-\frac{8311}{16\!\cdots\!75}a^{16}-\frac{720798015270217}{18\!\cdots\!25}a^{15}+\frac{37\!\cdots\!97}{74\!\cdots\!00}a^{14}+\frac{34\!\cdots\!13}{37\!\cdots\!50}a^{13}-\frac{66\!\cdots\!54}{18\!\cdots\!25}a^{12}+\frac{78\!\cdots\!71}{37\!\cdots\!50}a^{11}-\frac{65\!\cdots\!79}{37\!\cdots\!50}a^{10}+\frac{24\!\cdots\!08}{10\!\cdots\!25}a^{9}+\frac{64\!\cdots\!07}{18\!\cdots\!25}a^{8}-\frac{19\!\cdots\!99}{74\!\cdots\!10}a^{7}+\frac{32\!\cdots\!64}{18\!\cdots\!25}a^{6}-\frac{51\!\cdots\!41}{13\!\cdots\!20}a^{5}+\frac{20\!\cdots\!47}{33\!\cdots\!50}a^{4}+\frac{60\!\cdots\!88}{18\!\cdots\!25}a^{3}+\frac{14\!\cdots\!67}{74\!\cdots\!00}a^{2}+\frac{43\!\cdots\!63}{18\!\cdots\!25}a+\frac{10\!\cdots\!87}{64\!\cdots\!25}$ 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

$C_{3}\times C_{3}$, which has order $9$ (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:   \( \frac{1222750509}{1867395407900} a^{17} - \frac{324045153}{933697703950} a^{16} + \frac{3885541669}{933697703950} a^{15} - \frac{4313647947}{1867395407900} a^{14} + \frac{731853796}{42440804725} a^{13} - \frac{44933378901}{466848851975} a^{12} + \frac{56834447929}{933697703950} a^{11} - \frac{111999544071}{933697703950} a^{10} - \frac{139578117417}{933697703950} a^{9} + \frac{296901944733}{466848851975} a^{8} + \frac{247370444569}{186739540790} a^{7} + \frac{351031277127}{933697703950} a^{6} + \frac{965174150681}{373479081580} a^{5} - \frac{1589418772851}{466848851975} a^{4} - \frac{890455147118}{466848851975} a^{3} + \frac{9391142641263}{1867395407900} a^{2} + \frac{682351786989}{84881609450} a + \frac{117266644651}{32196472550} \)  (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:   $\frac{580228162847119}{18\!\cdots\!25}a^{17}-\frac{33\!\cdots\!37}{37\!\cdots\!50}a^{16}+\frac{10\!\cdots\!61}{37\!\cdots\!50}a^{15}-\frac{22\!\cdots\!49}{37\!\cdots\!50}a^{14}+\frac{43\!\cdots\!33}{33\!\cdots\!50}a^{13}-\frac{12\!\cdots\!84}{18\!\cdots\!25}a^{12}+\frac{26\!\cdots\!83}{18\!\cdots\!25}a^{11}-\frac{32\!\cdots\!82}{18\!\cdots\!25}a^{10}+\frac{31\!\cdots\!97}{37\!\cdots\!50}a^{9}+\frac{82\!\cdots\!77}{18\!\cdots\!25}a^{8}-\frac{15\!\cdots\!21}{74\!\cdots\!61}a^{7}-\frac{34\!\cdots\!57}{37\!\cdots\!50}a^{6}+\frac{57\!\cdots\!63}{37\!\cdots\!05}a^{5}-\frac{16\!\cdots\!93}{37\!\cdots\!50}a^{4}+\frac{81\!\cdots\!73}{18\!\cdots\!25}a^{3}+\frac{10\!\cdots\!01}{37\!\cdots\!50}a^{2}-\frac{12\!\cdots\!89}{33\!\cdots\!50}a-\frac{40\!\cdots\!31}{12\!\cdots\!50}$, $\frac{71\!\cdots\!07}{74\!\cdots\!00}a^{17}-\frac{23\!\cdots\!82}{18\!\cdots\!25}a^{16}+\frac{12\!\cdots\!66}{18\!\cdots\!25}a^{15}-\frac{61\!\cdots\!71}{74\!\cdots\!00}a^{14}+\frac{10\!\cdots\!61}{33\!\cdots\!50}a^{13}-\frac{30\!\cdots\!43}{18\!\cdots\!25}a^{12}+\frac{76\!\cdots\!77}{37\!\cdots\!50}a^{11}-\frac{10\!\cdots\!53}{37\!\cdots\!50}a^{10}-\frac{11\!\cdots\!53}{18\!\cdots\!25}a^{9}+\frac{20\!\cdots\!24}{18\!\cdots\!25}a^{8}+\frac{82\!\cdots\!61}{74\!\cdots\!10}a^{7}-\frac{10\!\cdots\!32}{18\!\cdots\!25}a^{6}+\frac{61\!\cdots\!31}{14\!\cdots\!20}a^{5}-\frac{29\!\cdots\!11}{37\!\cdots\!50}a^{4}+\frac{45\!\cdots\!11}{18\!\cdots\!25}a^{3}+\frac{58\!\cdots\!19}{74\!\cdots\!00}a^{2}+\frac{72\!\cdots\!31}{16\!\cdots\!75}a-\frac{33\!\cdots\!36}{64\!\cdots\!25}$, $\frac{82\!\cdots\!86}{18\!\cdots\!25}a^{17}-\frac{12\!\cdots\!53}{37\!\cdots\!50}a^{16}+\frac{21\!\cdots\!43}{74\!\cdots\!00}a^{15}-\frac{38\!\cdots\!28}{18\!\cdots\!25}a^{14}+\frac{44\!\cdots\!57}{37\!\cdots\!50}a^{13}-\frac{50\!\cdots\!39}{74\!\cdots\!00}a^{12}+\frac{18\!\cdots\!39}{33\!\cdots\!50}a^{11}-\frac{33\!\cdots\!81}{37\!\cdots\!50}a^{10}-\frac{36\!\cdots\!57}{37\!\cdots\!50}a^{9}+\frac{14\!\cdots\!81}{30\!\cdots\!50}a^{8}+\frac{58\!\cdots\!91}{74\!\cdots\!10}a^{7}+\frac{92\!\cdots\!13}{10\!\cdots\!25}a^{6}+\frac{92\!\cdots\!15}{51\!\cdots\!18}a^{5}-\frac{48\!\cdots\!91}{18\!\cdots\!25}a^{4}-\frac{15\!\cdots\!73}{25\!\cdots\!00}a^{3}+\frac{14\!\cdots\!89}{37\!\cdots\!50}a^{2}+\frac{80\!\cdots\!82}{18\!\cdots\!25}a+\frac{34\!\cdots\!27}{25\!\cdots\!00}$, $\frac{77630421369429}{74\!\cdots\!61}a^{17}-\frac{575373736932354}{37\!\cdots\!05}a^{16}+\frac{542929962923801}{74\!\cdots\!61}a^{15}-\frac{36\!\cdots\!22}{37\!\cdots\!05}a^{14}+\frac{11\!\cdots\!29}{37\!\cdots\!05}a^{13}-\frac{13\!\cdots\!54}{74\!\cdots\!61}a^{12}+\frac{81\!\cdots\!52}{33\!\cdots\!55}a^{11}-\frac{10\!\cdots\!46}{37\!\cdots\!05}a^{10}-\frac{37\!\cdots\!41}{37\!\cdots\!05}a^{9}+\frac{91\!\cdots\!96}{67\!\cdots\!51}a^{8}+\frac{35\!\cdots\!04}{37\!\cdots\!05}a^{7}-\frac{43\!\cdots\!47}{37\!\cdots\!05}a^{6}+\frac{16\!\cdots\!57}{37\!\cdots\!05}a^{5}-\frac{35\!\cdots\!39}{37\!\cdots\!05}a^{4}+\frac{81\!\cdots\!56}{43\!\cdots\!33}a^{3}+\frac{41\!\cdots\!28}{37\!\cdots\!05}a^{2}+\frac{79\!\cdots\!41}{37\!\cdots\!05}a-\frac{39\!\cdots\!53}{12\!\cdots\!45}$, $\frac{41\!\cdots\!11}{74\!\cdots\!00}a^{17}-\frac{38\!\cdots\!99}{74\!\cdots\!00}a^{16}+\frac{28\!\cdots\!67}{74\!\cdots\!00}a^{15}-\frac{27\!\cdots\!93}{74\!\cdots\!00}a^{14}+\frac{12\!\cdots\!31}{74\!\cdots\!00}a^{13}-\frac{65\!\cdots\!91}{74\!\cdots\!00}a^{12}+\frac{32\!\cdots\!41}{37\!\cdots\!50}a^{11}-\frac{27\!\cdots\!27}{18\!\cdots\!25}a^{10}-\frac{30\!\cdots\!43}{37\!\cdots\!50}a^{9}+\frac{11\!\cdots\!67}{18\!\cdots\!25}a^{8}+\frac{41\!\cdots\!83}{74\!\cdots\!10}a^{7}+\frac{15\!\cdots\!89}{18\!\cdots\!25}a^{6}+\frac{16\!\cdots\!03}{14\!\cdots\!20}a^{5}-\frac{16\!\cdots\!21}{74\!\cdots\!00}a^{4}-\frac{86\!\cdots\!33}{74\!\cdots\!00}a^{3}+\frac{33\!\cdots\!97}{74\!\cdots\!00}a^{2}+\frac{37\!\cdots\!47}{74\!\cdots\!00}a+\frac{28\!\cdots\!79}{15\!\cdots\!00}$, $\frac{95277113965179}{37\!\cdots\!50}a^{17}-\frac{11\!\cdots\!37}{74\!\cdots\!00}a^{16}+\frac{338967510692443}{43\!\cdots\!00}a^{15}-\frac{20\!\cdots\!56}{18\!\cdots\!25}a^{14}+\frac{22\!\cdots\!43}{74\!\cdots\!00}a^{13}-\frac{54\!\cdots\!63}{74\!\cdots\!00}a^{12}+\frac{56\!\cdots\!34}{18\!\cdots\!25}a^{11}-\frac{30\!\cdots\!47}{33\!\cdots\!50}a^{10}+\frac{43\!\cdots\!83}{18\!\cdots\!25}a^{9}+\frac{35\!\cdots\!96}{18\!\cdots\!25}a^{8}-\frac{15\!\cdots\!33}{43\!\cdots\!33}a^{7}+\frac{483912990035877}{21\!\cdots\!50}a^{6}+\frac{85\!\cdots\!97}{43\!\cdots\!30}a^{5}-\frac{96\!\cdots\!73}{74\!\cdots\!00}a^{4}-\frac{13\!\cdots\!59}{74\!\cdots\!00}a^{3}+\frac{26\!\cdots\!99}{18\!\cdots\!25}a^{2}+\frac{12\!\cdots\!41}{74\!\cdots\!00}a+\frac{85\!\cdots\!49}{25\!\cdots\!00}$, $\frac{38\!\cdots\!43}{74\!\cdots\!00}a^{17}+\frac{862414094096017}{18\!\cdots\!25}a^{16}+\frac{23\!\cdots\!61}{74\!\cdots\!00}a^{15}+\frac{22\!\cdots\!51}{74\!\cdots\!00}a^{14}+\frac{46\!\cdots\!79}{37\!\cdots\!50}a^{13}-\frac{24\!\cdots\!99}{43\!\cdots\!00}a^{12}-\frac{10\!\cdots\!56}{18\!\cdots\!25}a^{11}-\frac{13\!\cdots\!81}{18\!\cdots\!25}a^{10}-\frac{50\!\cdots\!62}{18\!\cdots\!25}a^{9}+\frac{12\!\cdots\!37}{37\!\cdots\!50}a^{8}+\frac{64\!\cdots\!19}{37\!\cdots\!05}a^{7}+\frac{76\!\cdots\!79}{37\!\cdots\!50}a^{6}+\frac{34\!\cdots\!77}{10\!\cdots\!36}a^{5}+\frac{16\!\cdots\!38}{18\!\cdots\!25}a^{4}-\frac{12\!\cdots\!81}{25\!\cdots\!00}a^{3}-\frac{93\!\cdots\!39}{67\!\cdots\!00}a^{2}+\frac{11\!\cdots\!03}{37\!\cdots\!50}a+\frac{38\!\cdots\!29}{25\!\cdots\!00}$, $\frac{112476607126144}{18\!\cdots\!25}a^{17}+\frac{398197604012261}{74\!\cdots\!00}a^{16}+\frac{41\!\cdots\!17}{74\!\cdots\!00}a^{15}-\frac{759054098511712}{18\!\cdots\!25}a^{14}+\frac{14\!\cdots\!91}{74\!\cdots\!00}a^{13}-\frac{86\!\cdots\!61}{74\!\cdots\!00}a^{12}-\frac{79\!\cdots\!52}{18\!\cdots\!25}a^{11}-\frac{19\!\cdots\!49}{37\!\cdots\!50}a^{10}+\frac{11\!\cdots\!71}{18\!\cdots\!25}a^{9}+\frac{17\!\cdots\!92}{18\!\cdots\!25}a^{8}+\frac{24\!\cdots\!89}{74\!\cdots\!61}a^{7}+\frac{24\!\cdots\!73}{37\!\cdots\!50}a^{6}+\frac{18\!\cdots\!13}{37\!\cdots\!05}a^{5}-\frac{10\!\cdots\!11}{74\!\cdots\!00}a^{4}-\frac{19\!\cdots\!33}{74\!\cdots\!00}a^{3}-\frac{13\!\cdots\!92}{18\!\cdots\!25}a^{2}-\frac{42\!\cdots\!63}{74\!\cdots\!00}a-\frac{60\!\cdots\!17}{25\!\cdots\!00}$ 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:  \( 11699771.704 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 11699771.704 \cdot 9}{6\cdot\sqrt{24962803242374579500132282368}}\cr\approx \mathstrut & 1.6952774490 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 + 24*x^14 - 132*x^13 + 13*x^12 - 120*x^11 - 348*x^10 + 864*x^9 + 2532*x^8 + 1536*x^7 + 4219*x^6 - 2976*x^5 - 5622*x^4 + 7200*x^3 + 16140*x^2 + 10440*x + 2523)
 
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^16 + 24*x^14 - 132*x^13 + 13*x^12 - 120*x^11 - 348*x^10 + 864*x^9 + 2532*x^8 + 1536*x^7 + 4219*x^6 - 2976*x^5 - 5622*x^4 + 7200*x^3 + 16140*x^2 + 10440*x + 2523, 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^16 + 24*x^14 - 132*x^13 + 13*x^12 - 120*x^11 - 348*x^10 + 864*x^9 + 2532*x^8 + 1536*x^7 + 4219*x^6 - 2976*x^5 - 5622*x^4 + 7200*x^3 + 16140*x^2 + 10440*x + 2523);
 
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^16 + 24*x^14 - 132*x^13 + 13*x^12 - 120*x^11 - 348*x^10 + 864*x^9 + 2532*x^8 + 1536*x^7 + 4219*x^6 - 2976*x^5 - 5622*x^4 + 7200*x^3 + 16140*x^2 + 10440*x + 2523);
 
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

$C_3:S_3$ (as 18T4):

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 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.31212.1 x3, 3.1.31212.2 x3, 3.1.108.1 x3, 3.1.867.1 x3, 6.0.2922566832.1, 6.0.2922566832.2, 6.0.34992.1, 6.0.2255067.2, 9.1.91219155960384.3 x9

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: 9.1.91219155960384.3
Minimal sibling: 9.1.91219155960384.3

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 ${\href{/padicField/5.2.0.1}{2} }^{9}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ R ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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 2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
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 3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
\(17\) Copy content Toggle raw display 17.6.4.1$x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$