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Properties

Label 18.0.450...000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-4.503\times 10^{29}$
Root discriminant \(44.40\)
Ramified primes $2,3,5$
Class number $108$ (GRH)
Class group [3, 6, 6] (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 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 216*x^12 - 729*x^11 + 3807*x^10 - 5414*x^9 - 423*x^8 + 17991*x^7 - 36621*x^6 + 11718*x^5 + 99414*x^4 - 188676*x^3 + 121500*x^2 + 11016*x + 1156)
 
gp: K = bnfinit(y^18 - 9*y^17 + 45*y^16 - 126*y^15 + 189*y^14 - 27*y^13 - 216*y^12 - 729*y^11 + 3807*y^10 - 5414*y^9 - 423*y^8 + 17991*y^7 - 36621*y^6 + 11718*y^5 + 99414*y^4 - 188676*y^3 + 121500*y^2 + 11016*y + 1156, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 216*x^12 - 729*x^11 + 3807*x^10 - 5414*x^9 - 423*x^8 + 17991*x^7 - 36621*x^6 + 11718*x^5 + 99414*x^4 - 188676*x^3 + 121500*x^2 + 11016*x + 1156);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 216*x^12 - 729*x^11 + 3807*x^10 - 5414*x^9 - 423*x^8 + 17991*x^7 - 36621*x^6 + 11718*x^5 + 99414*x^4 - 188676*x^3 + 121500*x^2 + 11016*x + 1156)
 

\( x^{18} - 9 x^{17} + 45 x^{16} - 126 x^{15} + 189 x^{14} - 27 x^{13} - 216 x^{12} - 729 x^{11} + \cdots + 1156 \) 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:   \(-450283905890997363000000000000\) \(\medspace = -\,2^{12}\cdot 3^{37}\cdot 5^{12}\) 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:  \(44.40\)
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:  $2^{2/3}3^{37/18}5^{2/3}\approx 44.40336798307826$
Ramified primes:   \(2\), \(3\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{7}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{45}a^{9}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{7}{45}$, $\frac{1}{45}a^{10}+\frac{1}{5}a^{5}+\frac{4}{9}a-\frac{1}{5}$, $\frac{1}{45}a^{11}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{11}{45}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{450}a^{12}+\frac{2}{225}a^{11}-\frac{2}{225}a^{10}-\frac{1}{90}a^{9}-\frac{2}{25}a^{7}-\frac{1}{50}a^{6}-\frac{7}{25}a^{5}-\frac{1}{5}a^{4}-\frac{1}{18}a^{3}-\frac{77}{225}a^{2}-\frac{13}{225}a-\frac{77}{225}$, $\frac{1}{1350}a^{13}-\frac{1}{1350}a^{12}+\frac{1}{225}a^{11}+\frac{1}{270}a^{10}-\frac{1}{270}a^{9}+\frac{1}{25}a^{8}-\frac{1}{150}a^{7}+\frac{11}{150}a^{6}+\frac{1}{5}a^{5}-\frac{77}{270}a^{4}-\frac{659}{1350}a^{3}+\frac{74}{225}a^{2}+\frac{203}{675}a-\frac{64}{135}$, $\frac{1}{1350}a^{14}-\frac{1}{1350}a^{12}-\frac{13}{1350}a^{11}-\frac{1}{225}a^{10}-\frac{11}{1350}a^{9}+\frac{1}{30}a^{8}+\frac{2}{75}a^{7}-\frac{13}{150}a^{6}+\frac{641}{1350}a^{5}+\frac{17}{75}a^{4}+\frac{41}{270}a^{3}+\frac{77}{675}a^{2}-\frac{68}{225}a-\frac{83}{675}$, $\frac{1}{1350}a^{15}+\frac{1}{1350}a^{12}-\frac{1}{225}a^{10}-\frac{1}{270}a^{9}+\frac{1}{15}a^{8}-\frac{7}{75}a^{7}+\frac{13}{270}a^{6}+\frac{32}{75}a^{5}-\frac{2}{15}a^{4}-\frac{61}{135}a^{3}+\frac{2}{75}a^{2}+\frac{8}{45}a+\frac{62}{135}$, $\frac{1}{83700}a^{16}+\frac{1}{20925}a^{15}+\frac{23}{83700}a^{14}+\frac{1}{83700}a^{13}+\frac{1}{1674}a^{12}-\frac{629}{83700}a^{11}-\frac{623}{83700}a^{10}-\frac{8}{2325}a^{9}-\frac{203}{3100}a^{8}-\frac{1337}{16740}a^{7}+\frac{191}{20925}a^{6}+\frac{34363}{83700}a^{5}-\frac{6268}{20925}a^{4}+\frac{6233}{41850}a^{3}-\frac{763}{8370}a^{2}+\frac{4328}{20925}a+\frac{11}{2325}$, $\frac{1}{21\!\cdots\!00}a^{17}-\frac{59536140920819}{10\!\cdots\!50}a^{16}-\frac{356602764606103}{24\!\cdots\!00}a^{15}-\frac{21177735745259}{24\!\cdots\!00}a^{14}+\frac{27\!\cdots\!11}{10\!\cdots\!50}a^{13}+\frac{21\!\cdots\!99}{21\!\cdots\!00}a^{12}+\frac{24\!\cdots\!69}{21\!\cdots\!00}a^{11}-\frac{15\!\cdots\!63}{18\!\cdots\!75}a^{10}+\frac{57\!\cdots\!91}{21\!\cdots\!00}a^{9}+\frac{84\!\cdots\!43}{21\!\cdots\!00}a^{8}+\frac{44\!\cdots\!64}{54\!\cdots\!25}a^{7}+\frac{77\!\cdots\!33}{80\!\cdots\!00}a^{6}-\frac{13\!\cdots\!13}{20\!\cdots\!75}a^{5}+\frac{20\!\cdots\!36}{54\!\cdots\!25}a^{4}-\frac{13\!\cdots\!82}{54\!\cdots\!25}a^{3}-\frac{637242093008258}{70\!\cdots\!39}a^{2}+\frac{15\!\cdots\!73}{18\!\cdots\!75}a-\frac{66\!\cdots\!31}{32\!\cdots\!25}$ 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

$C_{3}\times C_{6}\times C_{6}$, which has order $108$ (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:   \( -\frac{16580722975945}{218223796955517909} a^{17} + \frac{616288653199925}{872895187822071636} a^{16} - \frac{1549449380180957}{436447593911035818} a^{15} + \frac{8822454428403755}{872895187822071636} a^{14} - \frac{13264872400894195}{872895187822071636} a^{13} + \frac{543259332915545}{218223796955517909} a^{12} + \frac{16120294174521395}{872895187822071636} a^{11} + \frac{49458692589293621}{872895187822071636} a^{10} - \frac{132822660357257615}{436447593911035818} a^{9} + \frac{379599875740379095}{872895187822071636} a^{8} + \frac{24401006559029995}{872895187822071636} a^{7} - \frac{617704859030346235}{436447593911035818} a^{6} + \frac{2629026047845200643}{872895187822071636} a^{5} - \frac{218289606409466330}{218223796955517909} a^{4} - \frac{3459436774271394085}{436447593911035818} a^{3} + \frac{6591770227628359775}{436447593911035818} a^{2} - \frac{2089576757630355020}{218223796955517909} a + \frac{54616452757346}{414086901243867} \)  (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:   $\frac{20932347410989}{72\!\cdots\!30}a^{17}-\frac{30\!\cdots\!89}{10\!\cdots\!50}a^{16}+\frac{78\!\cdots\!72}{54\!\cdots\!25}a^{15}-\frac{46\!\cdots\!81}{10\!\cdots\!45}a^{14}+\frac{15\!\cdots\!61}{21\!\cdots\!90}a^{13}-\frac{27\!\cdots\!77}{10\!\cdots\!45}a^{12}-\frac{43\!\cdots\!51}{54\!\cdots\!25}a^{11}-\frac{64\!\cdots\!93}{36\!\cdots\!50}a^{10}+\frac{14\!\cdots\!06}{12\!\cdots\!05}a^{9}-\frac{73\!\cdots\!41}{36\!\cdots\!15}a^{8}+\frac{67\!\cdots\!23}{21\!\cdots\!90}a^{7}+\frac{32\!\cdots\!21}{54\!\cdots\!25}a^{6}-\frac{14\!\cdots\!71}{10\!\cdots\!50}a^{5}+\frac{16\!\cdots\!70}{21\!\cdots\!09}a^{4}+\frac{32\!\cdots\!48}{10\!\cdots\!45}a^{3}-\frac{15\!\cdots\!81}{21\!\cdots\!09}a^{2}+\frac{99\!\cdots\!19}{18\!\cdots\!75}a+\frac{388896603587753}{71\!\cdots\!65}$, $\frac{18812513506075}{43\!\cdots\!18}a^{17}-\frac{21\!\cdots\!42}{54\!\cdots\!25}a^{16}+\frac{851170432952231}{43\!\cdots\!18}a^{15}-\frac{60\!\cdots\!69}{10\!\cdots\!45}a^{14}+\frac{65\!\cdots\!63}{72\!\cdots\!30}a^{13}-\frac{127672504144048}{40\!\cdots\!35}a^{12}-\frac{13\!\cdots\!86}{18\!\cdots\!75}a^{11}-\frac{31\!\cdots\!51}{10\!\cdots\!50}a^{10}+\frac{17\!\cdots\!38}{10\!\cdots\!45}a^{9}-\frac{28\!\cdots\!08}{10\!\cdots\!45}a^{8}+\frac{94\!\cdots\!61}{21\!\cdots\!90}a^{7}+\frac{41\!\cdots\!18}{54\!\cdots\!25}a^{6}-\frac{17\!\cdots\!59}{10\!\cdots\!50}a^{5}+\frac{36\!\cdots\!89}{46\!\cdots\!26}a^{4}+\frac{30\!\cdots\!73}{80\!\cdots\!70}a^{3}-\frac{32\!\cdots\!21}{36\!\cdots\!15}a^{2}+\frac{75\!\cdots\!19}{10\!\cdots\!45}a-\frac{16\!\cdots\!24}{32\!\cdots\!25}$, $\frac{13014558088057}{60\!\cdots\!25}a^{17}-\frac{42\!\cdots\!69}{21\!\cdots\!00}a^{16}+\frac{651564821100217}{72\!\cdots\!30}a^{15}-\frac{19\!\cdots\!07}{87\!\cdots\!36}a^{14}+\frac{14\!\cdots\!27}{72\!\cdots\!00}a^{13}+\frac{19\!\cdots\!79}{72\!\cdots\!30}a^{12}-\frac{20\!\cdots\!63}{43\!\cdots\!80}a^{11}-\frac{50\!\cdots\!03}{21\!\cdots\!00}a^{10}+\frac{24\!\cdots\!39}{36\!\cdots\!15}a^{9}-\frac{62\!\cdots\!79}{24\!\cdots\!00}a^{8}-\frac{30\!\cdots\!81}{43\!\cdots\!80}a^{7}+\frac{86\!\cdots\!91}{36\!\cdots\!15}a^{6}-\frac{17\!\cdots\!07}{21\!\cdots\!00}a^{5}+\frac{18\!\cdots\!63}{72\!\cdots\!30}a^{4}+\frac{51\!\cdots\!63}{18\!\cdots\!75}a^{3}-\frac{34\!\cdots\!39}{10\!\cdots\!50}a^{2}+\frac{14\!\cdots\!59}{54\!\cdots\!25}a+\frac{36\!\cdots\!83}{10\!\cdots\!75}$, $\frac{70102800686131}{36\!\cdots\!50}a^{17}-\frac{32\!\cdots\!57}{21\!\cdots\!00}a^{16}+\frac{36\!\cdots\!98}{54\!\cdots\!25}a^{15}-\frac{23254193961797}{15\!\cdots\!20}a^{14}+\frac{21\!\cdots\!43}{21\!\cdots\!00}a^{13}+\frac{15\!\cdots\!07}{36\!\cdots\!50}a^{12}-\frac{14\!\cdots\!21}{14\!\cdots\!60}a^{11}-\frac{24\!\cdots\!47}{72\!\cdots\!00}a^{10}+\frac{38\!\cdots\!71}{10\!\cdots\!45}a^{9}-\frac{10\!\cdots\!81}{72\!\cdots\!00}a^{8}-\frac{25\!\cdots\!27}{21\!\cdots\!00}a^{7}+\frac{47\!\cdots\!37}{10\!\cdots\!45}a^{6}-\frac{53\!\cdots\!07}{80\!\cdots\!00}a^{5}+\frac{40\!\cdots\!36}{10\!\cdots\!45}a^{4}+\frac{77\!\cdots\!83}{36\!\cdots\!50}a^{3}-\frac{12\!\cdots\!67}{36\!\cdots\!50}a^{2}+\frac{42\!\cdots\!54}{18\!\cdots\!75}a+\frac{48\!\cdots\!02}{32\!\cdots\!25}$, $\frac{113699694883274}{54\!\cdots\!25}a^{17}-\frac{52\!\cdots\!03}{21\!\cdots\!00}a^{16}+\frac{13\!\cdots\!83}{10\!\cdots\!50}a^{15}-\frac{28\!\cdots\!21}{72\!\cdots\!00}a^{14}+\frac{13\!\cdots\!67}{21\!\cdots\!00}a^{13}-\frac{65\!\cdots\!43}{21\!\cdots\!90}a^{12}-\frac{11\!\cdots\!71}{21\!\cdots\!00}a^{11}-\frac{65\!\cdots\!29}{24\!\cdots\!00}a^{10}+\frac{62\!\cdots\!17}{54\!\cdots\!25}a^{9}-\frac{33\!\cdots\!07}{21\!\cdots\!00}a^{8}-\frac{49\!\cdots\!91}{43\!\cdots\!80}a^{7}+\frac{22\!\cdots\!44}{54\!\cdots\!25}a^{6}-\frac{86\!\cdots\!53}{72\!\cdots\!00}a^{5}+\frac{64\!\cdots\!71}{10\!\cdots\!50}a^{4}+\frac{94\!\cdots\!88}{54\!\cdots\!25}a^{3}-\frac{41\!\cdots\!67}{10\!\cdots\!50}a^{2}+\frac{22\!\cdots\!26}{60\!\cdots\!25}a-\frac{15\!\cdots\!17}{64\!\cdots\!85}$, $\frac{618685898753113}{10\!\cdots\!50}a^{17}-\frac{12\!\cdots\!31}{24\!\cdots\!00}a^{16}+\frac{30\!\cdots\!37}{12\!\cdots\!50}a^{15}-\frac{15\!\cdots\!13}{21\!\cdots\!00}a^{14}+\frac{22\!\cdots\!77}{21\!\cdots\!00}a^{13}-\frac{78\!\cdots\!89}{10\!\cdots\!50}a^{12}-\frac{27\!\cdots\!97}{21\!\cdots\!00}a^{11}-\frac{94\!\cdots\!43}{21\!\cdots\!00}a^{10}+\frac{11\!\cdots\!67}{54\!\cdots\!25}a^{9}-\frac{62\!\cdots\!37}{21\!\cdots\!00}a^{8}-\frac{15\!\cdots\!33}{24\!\cdots\!00}a^{7}+\frac{20\!\cdots\!49}{20\!\cdots\!75}a^{6}-\frac{43\!\cdots\!47}{21\!\cdots\!00}a^{5}+\frac{33\!\cdots\!63}{54\!\cdots\!25}a^{4}+\frac{30\!\cdots\!48}{54\!\cdots\!25}a^{3}-\frac{11\!\cdots\!11}{10\!\cdots\!50}a^{2}+\frac{33\!\cdots\!51}{54\!\cdots\!25}a+\frac{12\!\cdots\!56}{32\!\cdots\!25}$, $\frac{322603739731933}{43\!\cdots\!80}a^{17}-\frac{73\!\cdots\!37}{21\!\cdots\!00}a^{16}+\frac{23\!\cdots\!27}{21\!\cdots\!00}a^{15}+\frac{26\!\cdots\!11}{10\!\cdots\!50}a^{14}-\frac{10\!\cdots\!07}{21\!\cdots\!00}a^{13}+\frac{24\!\cdots\!03}{21\!\cdots\!00}a^{12}+\frac{18\!\cdots\!09}{10\!\cdots\!50}a^{11}-\frac{12\!\cdots\!41}{21\!\cdots\!00}a^{10}-\frac{75\!\cdots\!87}{21\!\cdots\!00}a^{9}+\frac{11\!\cdots\!48}{54\!\cdots\!25}a^{8}-\frac{44\!\cdots\!03}{21\!\cdots\!00}a^{7}+\frac{93\!\cdots\!77}{21\!\cdots\!00}a^{6}+\frac{23\!\cdots\!41}{21\!\cdots\!00}a^{5}-\frac{23\!\cdots\!09}{10\!\cdots\!50}a^{4}+\frac{15\!\cdots\!89}{10\!\cdots\!50}a^{3}+\frac{22\!\cdots\!19}{35\!\cdots\!50}a^{2}-\frac{12\!\cdots\!01}{54\!\cdots\!25}a-\frac{26\!\cdots\!83}{32\!\cdots\!25}$, $\frac{78109384333}{10\!\cdots\!30}a^{17}-\frac{242492714687}{407323932721452}a^{16}+\frac{1362086689583}{509154915901815}a^{15}-\frac{12550185402211}{20\!\cdots\!60}a^{14}+\frac{12646209136217}{20\!\cdots\!60}a^{13}+\frac{6003521529667}{10\!\cdots\!30}a^{12}-\frac{16357800586819}{20\!\cdots\!60}a^{11}-\frac{28399617019967}{407323932721452}a^{10}+\frac{98943780907409}{509154915901815}a^{9}-\frac{292007102936771}{20\!\cdots\!60}a^{8}-\frac{447210112519109}{20\!\cdots\!60}a^{7}+\frac{528496642922611}{509154915901815}a^{6}-\frac{30\!\cdots\!13}{20\!\cdots\!60}a^{5}-\frac{610173757355222}{509154915901815}a^{4}+\frac{61\!\cdots\!23}{10\!\cdots\!30}a^{3}-\frac{13\!\cdots\!05}{203661966360726}a^{2}+\frac{157729312514}{101830983180363}a-\frac{348638392172}{5990057834139}$ 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:  \( 2396291.4248 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 2396291.4248 \cdot 108}{6\cdot\sqrt{450283905890997363000000000000}}\cr\approx \mathstrut & 0.98104231852 \end{aligned}\] (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 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 216*x^12 - 729*x^11 + 3807*x^10 - 5414*x^9 - 423*x^8 + 17991*x^7 - 36621*x^6 + 11718*x^5 + 99414*x^4 - 188676*x^3 + 121500*x^2 + 11016*x + 1156)
 
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 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 216*x^12 - 729*x^11 + 3807*x^10 - 5414*x^9 - 423*x^8 + 17991*x^7 - 36621*x^6 + 11718*x^5 + 99414*x^4 - 188676*x^3 + 121500*x^2 + 11016*x + 1156, 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 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 216*x^12 - 729*x^11 + 3807*x^10 - 5414*x^9 - 423*x^8 + 17991*x^7 - 36621*x^6 + 11718*x^5 + 99414*x^4 - 188676*x^3 + 121500*x^2 + 11016*x + 1156);
 
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 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 216*x^12 - 729*x^11 + 3807*x^10 - 5414*x^9 - 423*x^8 + 17991*x^7 - 36621*x^6 + 11718*x^5 + 99414*x^4 - 188676*x^3 + 121500*x^2 + 11016*x + 1156);
 
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

\(\Q(\sqrt{-3}) \), 3.1.24300.1 x3, 3.1.24300.2 x3, 3.1.108.1 x3, 3.1.6075.2 x3, 6.0.1771470000.3, 6.0.1771470000.4, 6.0.34992.1, 6.0.110716875.2, 9.1.387420489000000.21 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: 9.1.387420489000000.21
Minimal sibling: 9.1.387420489000000.21

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 R ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.1.0.1}{1} }^{18}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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 Deg $18$$18$$1$$37$
\(5\) Copy content Toggle raw display 5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$