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Properties

Label 18.0.911...000.4
Degree $18$
Signature $[0, 9]$
Discriminant $-9.118\times 10^{28}$
Root discriminant \(40.63\)
Ramified primes $2,3,5,7$
Class number $54$ (GRH)
Class group [3, 18] (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^16 - 4*x^15 + 72*x^14 + 162*x^13 + 371*x^12 + 198*x^11 - 972*x^10 + 2976*x^9 + 10161*x^8 + 40986*x^7 + 67349*x^6 + 63342*x^5 + 188904*x^4 + 273560*x^3 + 182700*x^2 + 147000*x + 122500)
 
gp: K = bnfinit(y^18 + 9*y^16 - 4*y^15 + 72*y^14 + 162*y^13 + 371*y^12 + 198*y^11 - 972*y^10 + 2976*y^9 + 10161*y^8 + 40986*y^7 + 67349*y^6 + 63342*y^5 + 188904*y^4 + 273560*y^3 + 182700*y^2 + 147000*y + 122500, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 9*x^16 - 4*x^15 + 72*x^14 + 162*x^13 + 371*x^12 + 198*x^11 - 972*x^10 + 2976*x^9 + 10161*x^8 + 40986*x^7 + 67349*x^6 + 63342*x^5 + 188904*x^4 + 273560*x^3 + 182700*x^2 + 147000*x + 122500);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 9*x^16 - 4*x^15 + 72*x^14 + 162*x^13 + 371*x^12 + 198*x^11 - 972*x^10 + 2976*x^9 + 10161*x^8 + 40986*x^7 + 67349*x^6 + 63342*x^5 + 188904*x^4 + 273560*x^3 + 182700*x^2 + 147000*x + 122500)
 

\( x^{18} + 9 x^{16} - 4 x^{15} + 72 x^{14} + 162 x^{13} + 371 x^{12} + 198 x^{11} - 972 x^{10} + \cdots + 122500 \) 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:   \(-91176404162771495736000000000\) \(\medspace = -\,2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 7^{9}\) 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:  \(40.63\)
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:  $2^{2/3}3^{4/3}5^{1/2}7^{1/2}\approx 40.63332472677574$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(\Q(\sqrt{-35}) \)
$\card{ \Gal(K/\Q) }$:  $18$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{10}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{5}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{2}{5}a$, $\frac{1}{70}a^{10}+\frac{1}{70}a^{9}+\frac{2}{7}a^{8}+\frac{3}{14}a^{7}-\frac{1}{35}a^{6}-\frac{17}{70}a^{5}+\frac{1}{14}a^{4}-\frac{2}{7}a^{3}-\frac{12}{35}a^{2}-\frac{1}{5}a$, $\frac{1}{70}a^{11}-\frac{1}{35}a^{9}+\frac{3}{7}a^{8}-\frac{17}{70}a^{7}+\frac{2}{7}a^{6}-\frac{3}{35}a^{5}+\frac{1}{7}a^{4}+\frac{31}{70}a^{3}+\frac{1}{7}a^{2}+\frac{2}{5}a$, $\frac{1}{70}a^{12}-\frac{3}{70}a^{9}-\frac{6}{35}a^{8}-\frac{2}{7}a^{7}+\frac{5}{14}a^{6}-\frac{12}{35}a^{5}+\frac{3}{35}a^{4}+\frac{1}{14}a^{3}-\frac{2}{7}a^{2}-\frac{2}{5}a$, $\frac{1}{70}a^{13}-\frac{1}{35}a^{9}+\frac{1}{14}a^{8}+\frac{1}{14}a^{6}+\frac{11}{70}a^{5}-\frac{3}{14}a^{4}+\frac{5}{14}a^{3}-\frac{3}{7}a^{2}$, $\frac{1}{700}a^{14}+\frac{3}{700}a^{13}-\frac{3}{700}a^{12}+\frac{1}{175}a^{11}-\frac{3}{700}a^{10}-\frac{1}{700}a^{9}-\frac{339}{700}a^{8}-\frac{22}{175}a^{7}+\frac{33}{700}a^{6}+\frac{13}{700}a^{5}-\frac{3}{700}a^{4}-\frac{53}{350}a^{3}+\frac{61}{175}a^{2}-\frac{1}{10}a$, $\frac{1}{3500}a^{15}-\frac{1}{1750}a^{14}-\frac{2}{875}a^{13}+\frac{19}{3500}a^{12}-\frac{3}{3500}a^{11}+\frac{6}{875}a^{10}-\frac{26}{875}a^{9}-\frac{149}{500}a^{8}+\frac{283}{3500}a^{7}-\frac{561}{1750}a^{6}-\frac{377}{875}a^{5}+\frac{9}{3500}a^{4}+\frac{361}{875}a^{3}-\frac{23}{350}a^{2}-\frac{1}{10}a$, $\frac{1}{49808500}a^{16}-\frac{363}{9961700}a^{15}-\frac{32657}{49808500}a^{14}-\frac{45401}{24904250}a^{13}+\frac{1331}{284620}a^{12}+\frac{47613}{49808500}a^{11}+\frac{224009}{49808500}a^{10}-\frac{677583}{24904250}a^{9}-\frac{19265133}{49808500}a^{8}-\frac{201393}{7115500}a^{7}+\frac{16101753}{49808500}a^{6}-\frac{1872331}{24904250}a^{5}+\frac{2857313}{12452125}a^{4}+\frac{41663}{254125}a^{3}-\frac{1442}{50825}a^{2}-\frac{2726}{10165}a-\frac{9}{2033}$, $\frac{1}{12\!\cdots\!00}a^{17}+\frac{15\!\cdots\!71}{12\!\cdots\!00}a^{16}-\frac{42\!\cdots\!73}{31\!\cdots\!25}a^{15}+\frac{57\!\cdots\!41}{12\!\cdots\!00}a^{14}-\frac{17\!\cdots\!68}{31\!\cdots\!25}a^{13}+\frac{31\!\cdots\!43}{12\!\cdots\!00}a^{12}-\frac{16\!\cdots\!92}{31\!\cdots\!25}a^{11}+\frac{72\!\cdots\!09}{18\!\cdots\!00}a^{10}+\frac{58\!\cdots\!54}{31\!\cdots\!25}a^{9}-\frac{20\!\cdots\!49}{12\!\cdots\!00}a^{8}+\frac{31\!\cdots\!71}{63\!\cdots\!50}a^{7}+\frac{16\!\cdots\!93}{18\!\cdots\!00}a^{6}+\frac{13\!\cdots\!11}{13\!\cdots\!00}a^{5}-\frac{25\!\cdots\!56}{66\!\cdots\!95}a^{4}-\frac{16\!\cdots\!53}{90\!\cdots\!50}a^{3}-\frac{84\!\cdots\!07}{18\!\cdots\!50}a^{2}+\frac{10\!\cdots\!76}{25\!\cdots\!45}a-\frac{76\!\cdots\!29}{51\!\cdots\!89}$ 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:  $2$, $5$, $7$

Class group and class number

$C_{3}\times C_{18}$, which has order $54$ (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:   $\frac{75\!\cdots\!63}{12\!\cdots\!00}a^{17}-\frac{61\!\cdots\!73}{13\!\cdots\!00}a^{16}+\frac{40\!\cdots\!91}{63\!\cdots\!50}a^{15}-\frac{50\!\cdots\!81}{63\!\cdots\!50}a^{14}+\frac{69\!\cdots\!11}{12\!\cdots\!00}a^{13}+\frac{13\!\cdots\!29}{31\!\cdots\!25}a^{12}+\frac{77\!\cdots\!12}{31\!\cdots\!25}a^{11}-\frac{14\!\cdots\!39}{31\!\cdots\!25}a^{10}-\frac{42\!\cdots\!61}{12\!\cdots\!00}a^{9}+\frac{55\!\cdots\!37}{31\!\cdots\!25}a^{8}+\frac{28\!\cdots\!99}{63\!\cdots\!50}a^{7}+\frac{70\!\cdots\!52}{31\!\cdots\!25}a^{6}+\frac{89\!\cdots\!53}{31\!\cdots\!25}a^{5}+\frac{41\!\cdots\!11}{12\!\cdots\!00}a^{4}+\frac{12\!\cdots\!09}{12\!\cdots\!50}a^{3}+\frac{16\!\cdots\!39}{18\!\cdots\!50}a^{2}+\frac{41\!\cdots\!93}{51\!\cdots\!90}a+\frac{77\!\cdots\!98}{51\!\cdots\!89}$, $\frac{43\!\cdots\!21}{25\!\cdots\!00}a^{17}-\frac{29\!\cdots\!91}{12\!\cdots\!50}a^{16}+\frac{49\!\cdots\!57}{25\!\cdots\!00}a^{15}-\frac{88\!\cdots\!01}{25\!\cdots\!00}a^{14}+\frac{24\!\cdots\!03}{13\!\cdots\!00}a^{13}+\frac{12\!\cdots\!69}{25\!\cdots\!00}a^{12}+\frac{18\!\cdots\!11}{25\!\cdots\!00}a^{11}-\frac{17\!\cdots\!03}{25\!\cdots\!00}a^{10}-\frac{85\!\cdots\!37}{25\!\cdots\!00}a^{9}+\frac{12\!\cdots\!03}{25\!\cdots\!00}a^{8}+\frac{13\!\cdots\!93}{13\!\cdots\!00}a^{7}+\frac{14\!\cdots\!39}{25\!\cdots\!00}a^{6}+\frac{51\!\cdots\!87}{12\!\cdots\!50}a^{5}+\frac{41\!\cdots\!61}{50\!\cdots\!20}a^{4}+\frac{78\!\cdots\!91}{36\!\cdots\!30}a^{3}+\frac{12\!\cdots\!83}{72\!\cdots\!46}a^{2}+\frac{66\!\cdots\!99}{51\!\cdots\!90}a+\frac{77\!\cdots\!39}{51\!\cdots\!89}$, $\frac{19\!\cdots\!27}{98\!\cdots\!00}a^{17}+\frac{27\!\cdots\!67}{98\!\cdots\!00}a^{16}+\frac{48\!\cdots\!43}{28\!\cdots\!00}a^{15}+\frac{15\!\cdots\!19}{98\!\cdots\!00}a^{14}+\frac{31\!\cdots\!16}{24\!\cdots\!25}a^{13}+\frac{24\!\cdots\!91}{49\!\cdots\!50}a^{12}+\frac{11\!\cdots\!57}{98\!\cdots\!00}a^{11}+\frac{11\!\cdots\!67}{98\!\cdots\!00}a^{10}-\frac{41\!\cdots\!91}{24\!\cdots\!25}a^{9}+\frac{11\!\cdots\!32}{49\!\cdots\!25}a^{8}+\frac{27\!\cdots\!01}{98\!\cdots\!00}a^{7}+\frac{10\!\cdots\!89}{98\!\cdots\!00}a^{6}+\frac{33\!\cdots\!89}{14\!\cdots\!00}a^{5}+\frac{27\!\cdots\!81}{98\!\cdots\!00}a^{4}+\frac{29\!\cdots\!03}{70\!\cdots\!50}a^{3}+\frac{61\!\cdots\!44}{70\!\cdots\!75}a^{2}+\frac{40\!\cdots\!33}{40\!\cdots\!10}a+\frac{17\!\cdots\!51}{40\!\cdots\!01}$, $\frac{96\!\cdots\!01}{98\!\cdots\!00}a^{17}+\frac{19\!\cdots\!77}{39\!\cdots\!80}a^{16}+\frac{48\!\cdots\!89}{51\!\cdots\!00}a^{15}-\frac{10\!\cdots\!43}{98\!\cdots\!00}a^{14}+\frac{17\!\cdots\!54}{24\!\cdots\!25}a^{13}+\frac{17\!\cdots\!47}{98\!\cdots\!50}a^{12}+\frac{18\!\cdots\!99}{39\!\cdots\!80}a^{11}+\frac{30\!\cdots\!41}{98\!\cdots\!00}a^{10}-\frac{10\!\cdots\!41}{98\!\cdots\!45}a^{9}+\frac{12\!\cdots\!18}{98\!\cdots\!45}a^{8}+\frac{97\!\cdots\!67}{98\!\cdots\!00}a^{7}+\frac{24\!\cdots\!53}{51\!\cdots\!00}a^{6}+\frac{91\!\cdots\!93}{98\!\cdots\!00}a^{5}+\frac{18\!\cdots\!11}{19\!\cdots\!00}a^{4}+\frac{96\!\cdots\!33}{70\!\cdots\!50}a^{3}+\frac{20\!\cdots\!52}{14\!\cdots\!35}a^{2}+\frac{35\!\cdots\!19}{80\!\cdots\!02}a-\frac{84\!\cdots\!70}{40\!\cdots\!01}$, $\frac{85\!\cdots\!09}{12\!\cdots\!00}a^{17}+\frac{11\!\cdots\!67}{31\!\cdots\!25}a^{16}+\frac{82\!\cdots\!33}{12\!\cdots\!05}a^{15}-\frac{20\!\cdots\!73}{10\!\cdots\!44}a^{14}+\frac{19\!\cdots\!47}{36\!\cdots\!00}a^{13}+\frac{68\!\cdots\!77}{63\!\cdots\!50}a^{12}+\frac{11\!\cdots\!54}{31\!\cdots\!25}a^{11}+\frac{38\!\cdots\!17}{25\!\cdots\!00}a^{10}-\frac{86\!\cdots\!07}{12\!\cdots\!00}a^{9}+\frac{16\!\cdots\!61}{12\!\cdots\!50}a^{8}+\frac{21\!\cdots\!12}{31\!\cdots\!25}a^{7}+\frac{82\!\cdots\!03}{25\!\cdots\!00}a^{6}+\frac{18\!\cdots\!02}{31\!\cdots\!25}a^{5}+\frac{82\!\cdots\!48}{90\!\cdots\!75}a^{4}+\frac{56\!\cdots\!92}{45\!\cdots\!75}a^{3}+\frac{11\!\cdots\!86}{18\!\cdots\!15}a^{2}+\frac{17\!\cdots\!87}{25\!\cdots\!45}a+\frac{92\!\cdots\!52}{51\!\cdots\!89}$, $\frac{89\!\cdots\!53}{63\!\cdots\!50}a^{17}+\frac{74\!\cdots\!01}{18\!\cdots\!00}a^{16}+\frac{12\!\cdots\!63}{12\!\cdots\!00}a^{15}+\frac{96\!\cdots\!81}{31\!\cdots\!25}a^{14}+\frac{62\!\cdots\!81}{12\!\cdots\!00}a^{13}+\frac{35\!\cdots\!69}{63\!\cdots\!50}a^{12}+\frac{62\!\cdots\!43}{72\!\cdots\!60}a^{11}+\frac{95\!\cdots\!81}{63\!\cdots\!50}a^{10}-\frac{14\!\cdots\!63}{50\!\cdots\!20}a^{9}+\frac{82\!\cdots\!62}{31\!\cdots\!25}a^{8}+\frac{34\!\cdots\!21}{12\!\cdots\!00}a^{7}+\frac{57\!\cdots\!11}{59\!\cdots\!50}a^{6}+\frac{28\!\cdots\!73}{12\!\cdots\!00}a^{5}+\frac{26\!\cdots\!67}{12\!\cdots\!00}a^{4}+\frac{13\!\cdots\!42}{45\!\cdots\!75}a^{3}+\frac{62\!\cdots\!53}{90\!\cdots\!75}a^{2}+\frac{36\!\cdots\!81}{51\!\cdots\!90}a+\frac{11\!\cdots\!64}{51\!\cdots\!89}$, $\frac{37\!\cdots\!57}{90\!\cdots\!50}a^{17}-\frac{95\!\cdots\!09}{12\!\cdots\!00}a^{16}+\frac{24\!\cdots\!17}{63\!\cdots\!50}a^{15}-\frac{13\!\cdots\!03}{12\!\cdots\!00}a^{14}+\frac{11\!\cdots\!17}{31\!\cdots\!25}a^{13}-\frac{53\!\cdots\!02}{45\!\cdots\!75}a^{12}+\frac{11\!\cdots\!63}{12\!\cdots\!50}a^{11}-\frac{50\!\cdots\!99}{12\!\cdots\!00}a^{10}-\frac{30\!\cdots\!02}{63\!\cdots\!25}a^{9}+\frac{20\!\cdots\!24}{16\!\cdots\!75}a^{8}+\frac{15\!\cdots\!31}{45\!\cdots\!75}a^{7}+\frac{14\!\cdots\!07}{12\!\cdots\!00}a^{6}-\frac{45\!\cdots\!23}{63\!\cdots\!50}a^{5}-\frac{44\!\cdots\!29}{12\!\cdots\!00}a^{4}-\frac{19\!\cdots\!12}{64\!\cdots\!25}a^{3}-\frac{88\!\cdots\!59}{36\!\cdots\!30}a^{2}-\frac{15\!\cdots\!89}{51\!\cdots\!90}a-\frac{49\!\cdots\!81}{51\!\cdots\!89}$, $\frac{23\!\cdots\!99}{50\!\cdots\!20}a^{17}+\frac{17\!\cdots\!29}{12\!\cdots\!00}a^{16}+\frac{50\!\cdots\!11}{18\!\cdots\!00}a^{15}+\frac{92\!\cdots\!42}{31\!\cdots\!25}a^{14}+\frac{87\!\cdots\!49}{66\!\cdots\!00}a^{13}+\frac{17\!\cdots\!93}{12\!\cdots\!00}a^{12}-\frac{96\!\cdots\!79}{12\!\cdots\!00}a^{11}+\frac{23\!\cdots\!07}{63\!\cdots\!50}a^{10}-\frac{15\!\cdots\!07}{12\!\cdots\!00}a^{9}+\frac{34\!\cdots\!47}{12\!\cdots\!00}a^{8}+\frac{25\!\cdots\!53}{66\!\cdots\!00}a^{7}+\frac{11\!\cdots\!29}{63\!\cdots\!50}a^{6}+\frac{29\!\cdots\!69}{90\!\cdots\!50}a^{5}-\frac{60\!\cdots\!61}{31\!\cdots\!25}a^{4}+\frac{20\!\cdots\!61}{18\!\cdots\!15}a^{3}+\frac{93\!\cdots\!21}{12\!\cdots\!25}a^{2}+\frac{13\!\cdots\!50}{51\!\cdots\!89}a+\frac{38\!\cdots\!47}{51\!\cdots\!89}$ 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:  \( 5817988.03207 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 5817988.03207 \cdot 54}{2\cdot\sqrt{91176404162771495736000000000}}\cr\approx \mathstrut & 7.93988427799 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 4*x^15 + 72*x^14 + 162*x^13 + 371*x^12 + 198*x^11 - 972*x^10 + 2976*x^9 + 10161*x^8 + 40986*x^7 + 67349*x^6 + 63342*x^5 + 188904*x^4 + 273560*x^3 + 182700*x^2 + 147000*x + 122500)
 
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^16 - 4*x^15 + 72*x^14 + 162*x^13 + 371*x^12 + 198*x^11 - 972*x^10 + 2976*x^9 + 10161*x^8 + 40986*x^7 + 67349*x^6 + 63342*x^5 + 188904*x^4 + 273560*x^3 + 182700*x^2 + 147000*x + 122500, 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^16 - 4*x^15 + 72*x^14 + 162*x^13 + 371*x^12 + 198*x^11 - 972*x^10 + 2976*x^9 + 10161*x^8 + 40986*x^7 + 67349*x^6 + 63342*x^5 + 188904*x^4 + 273560*x^3 + 182700*x^2 + 147000*x + 122500);
 
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^16 - 4*x^15 + 72*x^14 + 162*x^13 + 371*x^12 + 198*x^11 - 972*x^10 + 2976*x^9 + 10161*x^8 + 40986*x^7 + 67349*x^6 + 63342*x^5 + 188904*x^4 + 273560*x^3 + 182700*x^2 + 147000*x + 122500);
 
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

\(\Q(\sqrt{-35}) \), 3.1.140.1 x3, 3.1.11340.1 x3, 3.1.2835.1 x3, 3.1.11340.2 x3, 6.0.686000.1, 6.0.4500846000.2, 6.0.281302875.1, 6.0.4500846000.1, 9.1.51039593640000.10 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: 9.1.51039593640000.10
Minimal sibling: 9.1.51039593640000.10

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 R ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{9}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.2.0.1}{2} }^{9}$ ${\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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 3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
\(5\) Copy content Toggle raw display 5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
\(7\) Copy content Toggle raw display 7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$