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Properties

Label 18.0.519...867.3
Degree $18$
Signature $[0, 9]$
Discriminant $-5.192\times 10^{29}$
Root discriminant \(44.76\)
Ramified primes $3,107$
Class number $36$ (GRH)
Class group [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 - 3*x^16 - 36*x^15 + 171*x^14 - 312*x^13 + 616*x^12 - 2430*x^11 + 7284*x^10 - 19902*x^9 + 51336*x^8 - 111702*x^7 + 202515*x^6 - 314298*x^5 + 421431*x^4 - 437878*x^3 + 392661*x^2 - 281418*x + 110764)
 
gp: K = bnfinit(y^18 - 3*y^16 - 36*y^15 + 171*y^14 - 312*y^13 + 616*y^12 - 2430*y^11 + 7284*y^10 - 19902*y^9 + 51336*y^8 - 111702*y^7 + 202515*y^6 - 314298*y^5 + 421431*y^4 - 437878*y^3 + 392661*y^2 - 281418*y + 110764, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^16 - 36*x^15 + 171*x^14 - 312*x^13 + 616*x^12 - 2430*x^11 + 7284*x^10 - 19902*x^9 + 51336*x^8 - 111702*x^7 + 202515*x^6 - 314298*x^5 + 421431*x^4 - 437878*x^3 + 392661*x^2 - 281418*x + 110764);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^16 - 36*x^15 + 171*x^14 - 312*x^13 + 616*x^12 - 2430*x^11 + 7284*x^10 - 19902*x^9 + 51336*x^8 - 111702*x^7 + 202515*x^6 - 314298*x^5 + 421431*x^4 - 437878*x^3 + 392661*x^2 - 281418*x + 110764)
 

\( x^{18} - 3 x^{16} - 36 x^{15} + 171 x^{14} - 312 x^{13} + 616 x^{12} - 2430 x^{11} + 7284 x^{10} + \cdots + 110764 \) 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:   \(-519235183203048555634413069867\) \(\medspace = -\,3^{24}\cdot 107^{9}\) 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:  \(44.76\)
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:  $3^{4/3}107^{1/2}\approx 44.75623667824389$
Ramified primes:   \(3\), \(107\) 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:  \(\Q(\sqrt{-107}) \)
$\card{ \Gal(K/\Q) }$:  $18$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{16}a^{11}-\frac{1}{8}a^{9}+\frac{1}{16}a^{8}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{3}{16}a^{5}-\frac{3}{8}a^{3}+\frac{3}{16}a^{2}-\frac{3}{8}a+\frac{1}{4}$, $\frac{1}{16}a^{12}-\frac{1}{8}a^{10}+\frac{1}{16}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{3}{16}a^{6}+\frac{1}{8}a^{4}+\frac{3}{16}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{16}a^{13}+\frac{1}{16}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{16}a^{7}+\frac{3}{16}a^{4}+\frac{1}{8}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{6016}a^{14}-\frac{21}{3008}a^{13}-\frac{1}{47}a^{12}+\frac{79}{3008}a^{11}+\frac{11}{94}a^{10}-\frac{43}{752}a^{9}+\frac{39}{376}a^{8}-\frac{19}{376}a^{7}+\frac{295}{1504}a^{6}+\frac{629}{3008}a^{5}-\frac{49}{376}a^{4}+\frac{143}{752}a^{3}-\frac{1481}{6016}a^{2}+\frac{1389}{3008}a+\frac{125}{1504}$, $\frac{1}{42112}a^{15}+\frac{1}{42112}a^{14}-\frac{403}{21056}a^{13}+\frac{21}{3008}a^{12}-\frac{387}{21056}a^{11}+\frac{101}{1316}a^{10}+\frac{43}{658}a^{9}-\frac{269}{2632}a^{8}+\frac{505}{10528}a^{7}+\frac{1371}{21056}a^{6}+\frac{711}{21056}a^{5}-\frac{39}{376}a^{4}-\frac{9817}{42112}a^{3}+\frac{1}{6016}a^{2}+\frac{511}{3008}a+\frac{2367}{10528}$, $\frac{1}{360478720}a^{16}+\frac{2599}{360478720}a^{15}-\frac{127}{10299392}a^{14}+\frac{960713}{90119680}a^{13}+\frac{3814067}{180239360}a^{12}+\frac{1427619}{180239360}a^{11}+\frac{2362391}{22529920}a^{10}-\frac{149531}{4505984}a^{9}-\frac{8710811}{90119680}a^{8}-\frac{42386609}{180239360}a^{7}+\frac{40545823}{180239360}a^{6}+\frac{1240873}{180239360}a^{5}+\frac{59442471}{360478720}a^{4}-\frac{139459407}{360478720}a^{3}+\frac{101413}{219136}a^{2}+\frac{31944859}{180239360}a+\frac{17787617}{90119680}$, $\frac{1}{24\!\cdots\!60}a^{17}-\frac{172273463753}{38\!\cdots\!40}a^{16}-\frac{21\!\cdots\!81}{26\!\cdots\!40}a^{15}+\frac{13\!\cdots\!27}{24\!\cdots\!60}a^{14}+\frac{15\!\cdots\!01}{12\!\cdots\!80}a^{13}+\frac{18\!\cdots\!91}{61\!\cdots\!40}a^{12}+\frac{40\!\cdots\!17}{17\!\cdots\!40}a^{11}-\frac{21\!\cdots\!91}{96\!\cdots\!60}a^{10}-\frac{32\!\cdots\!73}{57\!\cdots\!20}a^{9}-\frac{85\!\cdots\!41}{17\!\cdots\!40}a^{8}+\frac{62\!\cdots\!11}{61\!\cdots\!40}a^{7}+\frac{15\!\cdots\!01}{77\!\cdots\!48}a^{6}-\frac{17\!\cdots\!27}{49\!\cdots\!72}a^{5}+\frac{19\!\cdots\!27}{27\!\cdots\!60}a^{4}+\frac{55\!\cdots\!81}{12\!\cdots\!80}a^{3}+\frac{90\!\cdots\!63}{24\!\cdots\!60}a^{2}-\frac{11\!\cdots\!21}{35\!\cdots\!48}a+\frac{20\!\cdots\!93}{61\!\cdots\!40}$ 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

$C_{6}\times C_{6}$, which has order $36$ (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:   $\frac{19\!\cdots\!39}{14\!\cdots\!80}a^{17}-\frac{13\!\cdots\!39}{20\!\cdots\!40}a^{16}-\frac{33\!\cdots\!23}{30\!\cdots\!40}a^{15}-\frac{40\!\cdots\!11}{72\!\cdots\!40}a^{14}+\frac{19\!\cdots\!29}{72\!\cdots\!40}a^{13}-\frac{18\!\cdots\!57}{72\!\cdots\!40}a^{12}+\frac{72\!\cdots\!03}{36\!\cdots\!20}a^{11}-\frac{26\!\cdots\!99}{90\!\cdots\!80}a^{10}+\frac{34\!\cdots\!31}{36\!\cdots\!20}a^{9}-\frac{13\!\cdots\!83}{72\!\cdots\!40}a^{8}+\frac{52\!\cdots\!49}{10\!\cdots\!20}a^{7}-\frac{15\!\cdots\!19}{14\!\cdots\!08}a^{6}+\frac{45\!\cdots\!69}{29\!\cdots\!16}a^{5}-\frac{27\!\cdots\!67}{14\!\cdots\!80}a^{4}+\frac{30\!\cdots\!33}{14\!\cdots\!80}a^{3}-\frac{62\!\cdots\!57}{36\!\cdots\!20}a^{2}+\frac{30\!\cdots\!09}{36\!\cdots\!52}a-\frac{66\!\cdots\!99}{18\!\cdots\!60}$, $\frac{14\!\cdots\!57}{12\!\cdots\!80}a^{17}+\frac{31\!\cdots\!09}{24\!\cdots\!60}a^{16}+\frac{26\!\cdots\!21}{52\!\cdots\!80}a^{15}-\frac{58\!\cdots\!67}{24\!\cdots\!60}a^{14}-\frac{12\!\cdots\!41}{30\!\cdots\!20}a^{13}-\frac{27\!\cdots\!73}{12\!\cdots\!80}a^{12}+\frac{52\!\cdots\!69}{12\!\cdots\!80}a^{11}-\frac{40\!\cdots\!57}{15\!\cdots\!60}a^{10}-\frac{22\!\cdots\!37}{30\!\cdots\!20}a^{9}-\frac{97\!\cdots\!33}{30\!\cdots\!20}a^{8}+\frac{20\!\cdots\!75}{24\!\cdots\!36}a^{7}-\frac{19\!\cdots\!89}{12\!\cdots\!80}a^{6}+\frac{28\!\cdots\!33}{61\!\cdots\!40}a^{5}-\frac{62\!\cdots\!97}{49\!\cdots\!72}a^{4}+\frac{43\!\cdots\!69}{24\!\cdots\!60}a^{3}-\frac{46\!\cdots\!33}{24\!\cdots\!60}a^{2}+\frac{32\!\cdots\!73}{12\!\cdots\!80}a-\frac{12\!\cdots\!89}{61\!\cdots\!40}$, $\frac{87\!\cdots\!51}{24\!\cdots\!60}a^{17}+\frac{44\!\cdots\!03}{24\!\cdots\!60}a^{16}-\frac{21\!\cdots\!47}{52\!\cdots\!80}a^{15}-\frac{21\!\cdots\!19}{12\!\cdots\!80}a^{14}+\frac{86\!\cdots\!81}{12\!\cdots\!80}a^{13}+\frac{55\!\cdots\!67}{12\!\cdots\!80}a^{12}-\frac{14\!\cdots\!33}{61\!\cdots\!40}a^{11}-\frac{11\!\cdots\!71}{15\!\cdots\!60}a^{10}+\frac{16\!\cdots\!79}{61\!\cdots\!40}a^{9}-\frac{26\!\cdots\!47}{12\!\cdots\!80}a^{8}+\frac{32\!\cdots\!27}{12\!\cdots\!80}a^{7}-\frac{12\!\cdots\!11}{24\!\cdots\!36}a^{6}+\frac{60\!\cdots\!21}{49\!\cdots\!72}a^{5}-\frac{12\!\cdots\!83}{24\!\cdots\!60}a^{4}+\frac{28\!\cdots\!37}{24\!\cdots\!60}a^{3}-\frac{10\!\cdots\!33}{61\!\cdots\!40}a^{2}+\frac{86\!\cdots\!41}{61\!\cdots\!84}a-\frac{72\!\cdots\!11}{30\!\cdots\!20}$, $\frac{16\!\cdots\!03}{12\!\cdots\!80}a^{17}-\frac{14\!\cdots\!31}{12\!\cdots\!80}a^{16}-\frac{53\!\cdots\!21}{26\!\cdots\!40}a^{15}-\frac{56\!\cdots\!21}{30\!\cdots\!20}a^{14}+\frac{47\!\cdots\!93}{61\!\cdots\!40}a^{13}-\frac{10\!\cdots\!19}{61\!\cdots\!40}a^{12}+\frac{13\!\cdots\!39}{77\!\cdots\!80}a^{11}-\frac{74\!\cdots\!33}{77\!\cdots\!80}a^{10}+\frac{10\!\cdots\!87}{30\!\cdots\!20}a^{9}-\frac{45\!\cdots\!11}{61\!\cdots\!40}a^{8}+\frac{17\!\cdots\!69}{82\!\cdots\!60}a^{7}-\frac{64\!\cdots\!81}{12\!\cdots\!68}a^{6}+\frac{25\!\cdots\!45}{24\!\cdots\!36}a^{5}-\frac{21\!\cdots\!89}{12\!\cdots\!80}a^{4}+\frac{28\!\cdots\!71}{12\!\cdots\!80}a^{3}-\frac{15\!\cdots\!63}{61\!\cdots\!40}a^{2}+\frac{12\!\cdots\!27}{61\!\cdots\!84}a-\frac{17\!\cdots\!01}{19\!\cdots\!20}$, $\frac{87\!\cdots\!53}{49\!\cdots\!72}a^{17}+\frac{58\!\cdots\!57}{12\!\cdots\!80}a^{16}-\frac{63\!\cdots\!49}{65\!\cdots\!60}a^{15}-\frac{32\!\cdots\!11}{49\!\cdots\!72}a^{14}+\frac{37\!\cdots\!69}{12\!\cdots\!80}a^{13}-\frac{10\!\cdots\!53}{30\!\cdots\!20}a^{12}+\frac{26\!\cdots\!61}{12\!\cdots\!80}a^{11}-\frac{31\!\cdots\!39}{11\!\cdots\!40}a^{10}+\frac{12\!\cdots\!77}{12\!\cdots\!68}a^{9}-\frac{29\!\cdots\!03}{12\!\cdots\!80}a^{8}+\frac{16\!\cdots\!31}{30\!\cdots\!20}a^{7}-\frac{62\!\cdots\!49}{61\!\cdots\!40}a^{6}+\frac{32\!\cdots\!89}{24\!\cdots\!60}a^{5}-\frac{12\!\cdots\!13}{12\!\cdots\!80}a^{4}+\frac{14\!\cdots\!73}{61\!\cdots\!40}a^{3}+\frac{80\!\cdots\!17}{49\!\cdots\!72}a^{2}-\frac{32\!\cdots\!89}{12\!\cdots\!80}a+\frac{84\!\cdots\!83}{61\!\cdots\!40}$, $\frac{10\!\cdots\!43}{45\!\cdots\!40}a^{17}-\frac{11\!\cdots\!17}{20\!\cdots\!40}a^{16}-\frac{39\!\cdots\!31}{30\!\cdots\!40}a^{15}-\frac{12\!\cdots\!13}{14\!\cdots\!80}a^{14}+\frac{15\!\cdots\!77}{36\!\cdots\!20}a^{13}-\frac{42\!\cdots\!97}{72\!\cdots\!40}a^{12}+\frac{61\!\cdots\!91}{72\!\cdots\!40}a^{11}-\frac{46\!\cdots\!53}{90\!\cdots\!80}a^{10}+\frac{14\!\cdots\!01}{90\!\cdots\!80}a^{9}-\frac{14\!\cdots\!79}{36\!\cdots\!20}a^{8}+\frac{20\!\cdots\!09}{20\!\cdots\!44}a^{7}-\frac{15\!\cdots\!41}{72\!\cdots\!40}a^{6}+\frac{25\!\cdots\!69}{72\!\cdots\!40}a^{5}-\frac{14\!\cdots\!41}{29\!\cdots\!16}a^{4}+\frac{78\!\cdots\!81}{14\!\cdots\!80}a^{3}-\frac{75\!\cdots\!67}{14\!\cdots\!80}a^{2}+\frac{25\!\cdots\!47}{72\!\cdots\!40}a-\frac{53\!\cdots\!71}{36\!\cdots\!20}$, $\frac{17\!\cdots\!31}{24\!\cdots\!60}a^{17}+\frac{63\!\cdots\!63}{12\!\cdots\!80}a^{16}-\frac{27\!\cdots\!77}{65\!\cdots\!60}a^{15}-\frac{11\!\cdots\!73}{24\!\cdots\!60}a^{14}-\frac{69\!\cdots\!01}{12\!\cdots\!80}a^{13}+\frac{20\!\cdots\!97}{30\!\cdots\!20}a^{12}-\frac{17\!\cdots\!29}{12\!\cdots\!80}a^{11}+\frac{12\!\cdots\!41}{77\!\cdots\!80}a^{10}-\frac{20\!\cdots\!81}{61\!\cdots\!40}a^{9}+\frac{19\!\cdots\!67}{12\!\cdots\!80}a^{8}-\frac{42\!\cdots\!71}{61\!\cdots\!84}a^{7}+\frac{11\!\cdots\!97}{61\!\cdots\!40}a^{6}-\frac{85\!\cdots\!37}{24\!\cdots\!60}a^{5}+\frac{11\!\cdots\!61}{24\!\cdots\!36}a^{4}-\frac{33\!\cdots\!01}{61\!\cdots\!40}a^{3}+\frac{11\!\cdots\!63}{24\!\cdots\!60}a^{2}-\frac{31\!\cdots\!43}{12\!\cdots\!80}a+\frac{41\!\cdots\!29}{61\!\cdots\!40}$, $\frac{25\!\cdots\!11}{49\!\cdots\!72}a^{17}-\frac{35\!\cdots\!89}{24\!\cdots\!60}a^{16}-\frac{90\!\cdots\!23}{52\!\cdots\!80}a^{15}-\frac{39\!\cdots\!15}{24\!\cdots\!36}a^{14}+\frac{11\!\cdots\!81}{12\!\cdots\!80}a^{13}-\frac{33\!\cdots\!79}{17\!\cdots\!40}a^{12}+\frac{14\!\cdots\!27}{61\!\cdots\!40}a^{11}-\frac{15\!\cdots\!39}{15\!\cdots\!60}a^{10}+\frac{67\!\cdots\!01}{17\!\cdots\!24}a^{9}-\frac{12\!\cdots\!47}{12\!\cdots\!80}a^{8}+\frac{30\!\cdots\!31}{12\!\cdots\!80}a^{7}-\frac{63\!\cdots\!47}{12\!\cdots\!80}a^{6}+\frac{21\!\cdots\!81}{24\!\cdots\!60}a^{5}-\frac{28\!\cdots\!19}{24\!\cdots\!60}a^{4}+\frac{43\!\cdots\!99}{35\!\cdots\!80}a^{3}-\frac{11\!\cdots\!09}{12\!\cdots\!68}a^{2}+\frac{11\!\cdots\!01}{30\!\cdots\!20}a-\frac{16\!\cdots\!37}{44\!\cdots\!60}$ 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:  \( 9603674.43615 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 9603674.43615 \cdot 36}{2\cdot\sqrt{519235183203048555634413069867}}\cr\approx \mathstrut & 3.66139591945 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 36*x^15 + 171*x^14 - 312*x^13 + 616*x^12 - 2430*x^11 + 7284*x^10 - 19902*x^9 + 51336*x^8 - 111702*x^7 + 202515*x^6 - 314298*x^5 + 421431*x^4 - 437878*x^3 + 392661*x^2 - 281418*x + 110764)
 
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 - 3*x^16 - 36*x^15 + 171*x^14 - 312*x^13 + 616*x^12 - 2430*x^11 + 7284*x^10 - 19902*x^9 + 51336*x^8 - 111702*x^7 + 202515*x^6 - 314298*x^5 + 421431*x^4 - 437878*x^3 + 392661*x^2 - 281418*x + 110764, 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 - 3*x^16 - 36*x^15 + 171*x^14 - 312*x^13 + 616*x^12 - 2430*x^11 + 7284*x^10 - 19902*x^9 + 51336*x^8 - 111702*x^7 + 202515*x^6 - 314298*x^5 + 421431*x^4 - 437878*x^3 + 392661*x^2 - 281418*x + 110764);
 
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 - 3*x^16 - 36*x^15 + 171*x^14 - 312*x^13 + 616*x^12 - 2430*x^11 + 7284*x^10 - 19902*x^9 + 51336*x^8 - 111702*x^7 + 202515*x^6 - 314298*x^5 + 421431*x^4 - 437878*x^3 + 392661*x^2 - 281418*x + 110764);
 
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

\(\Q(\sqrt{-107}) \), 3.1.8667.2 x3, 3.1.107.1 x3, 3.1.8667.3 x3, 3.1.8667.1 x3, 6.0.8037507123.3, 6.0.1225043.1, 6.0.8037507123.4, 6.0.8037507123.2, 9.1.69661074235041.6 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: 9.1.69661074235041.6
Minimal sibling: 9.1.69661074235041.6

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{9}$ R ${\href{/padicField/5.2.0.1}{2} }^{9}$ ${\href{/padicField/7.2.0.1}{2} }^{9}$ ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.2.0.1}{2} }^{9}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.1.0.1}{1} }^{18}$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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
\(3\) 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}$
\(107\) Copy content Toggle raw display 107.2.1.2$x^{2} + 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 107$$2$$1$$1$$C_2$$[\ ]_{2}$