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Properties

Label 18.0.706...416.2
Degree $18$
Signature $[0, 9]$
Discriminant $-7.066\times 10^{42}$
Root discriminant \(240.16\)
Ramified primes $2,3,7,19$
Class number $67995072$ (GRH)
Class group [2, 2, 228, 74556] (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 98*x^16 - 230*x^15 + 7698*x^14 - 12222*x^13 + 450829*x^12 - 294056*x^11 + 20316110*x^10 - 1036720*x^9 + 691813801*x^8 + 163570074*x^7 + 17189420781*x^6 + 4958427054*x^5 + 301709383024*x^4 + 51432868270*x^3 + 3384649707670*x^2 + 106026030264*x + 17934199170349)
 
gp: K = bnfinit(y^18 - 2*y^17 + 98*y^16 - 230*y^15 + 7698*y^14 - 12222*y^13 + 450829*y^12 - 294056*y^11 + 20316110*y^10 - 1036720*y^9 + 691813801*y^8 + 163570074*y^7 + 17189420781*y^6 + 4958427054*y^5 + 301709383024*y^4 + 51432868270*y^3 + 3384649707670*y^2 + 106026030264*y + 17934199170349, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 98*x^16 - 230*x^15 + 7698*x^14 - 12222*x^13 + 450829*x^12 - 294056*x^11 + 20316110*x^10 - 1036720*x^9 + 691813801*x^8 + 163570074*x^7 + 17189420781*x^6 + 4958427054*x^5 + 301709383024*x^4 + 51432868270*x^3 + 3384649707670*x^2 + 106026030264*x + 17934199170349);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + 98*x^16 - 230*x^15 + 7698*x^14 - 12222*x^13 + 450829*x^12 - 294056*x^11 + 20316110*x^10 - 1036720*x^9 + 691813801*x^8 + 163570074*x^7 + 17189420781*x^6 + 4958427054*x^5 + 301709383024*x^4 + 51432868270*x^3 + 3384649707670*x^2 + 106026030264*x + 17934199170349)
 

\( x^{18} - 2 x^{17} + 98 x^{16} - 230 x^{15} + 7698 x^{14} - 12222 x^{13} + 450829 x^{12} + \cdots + 17934199170349 \) 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:   \(-7065768916593110856047783889584316216508416\) \(\medspace = -\,2^{18}\cdot 3^{9}\cdot 7^{15}\cdot 19^{16}\) 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:  \(240.16\)
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:  $2\cdot 3^{1/2}7^{5/6}19^{8/9}\approx 240.16452477246145$
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:  \(\Q(\sqrt{-21}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1596=2^{2}\cdot 3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1596}(1,·)$, $\chi_{1596}(899,·)$, $\chi_{1596}(923,·)$, $\chi_{1596}(1261,·)$, $\chi_{1596}(1297,·)$, $\chi_{1596}(83,·)$, $\chi_{1596}(719,·)$, $\chi_{1596}(25,·)$, $\chi_{1596}(731,·)$, $\chi_{1596}(803,·)$, $\chi_{1596}(479,·)$, $\chi_{1596}(419,·)$, $\chi_{1596}(131,·)$, $\chi_{1596}(625,·)$, $\chi_{1596}(1453,·)$, $\chi_{1596}(1201,·)$, $\chi_{1596}(505,·)$, $\chi_{1596}(1213,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{256}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{8}+\frac{3}{7}a^{7}-\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{1}{7}a^{3}$, $\frac{1}{77}a^{10}-\frac{3}{77}a^{9}-\frac{2}{77}a^{8}-\frac{27}{77}a^{7}+\frac{26}{77}a^{6}-\frac{12}{77}a^{5}-\frac{30}{77}a^{4}-\frac{9}{77}a^{3}-\frac{4}{11}a^{2}+\frac{1}{11}a$, $\frac{1}{77}a^{11}+\frac{3}{7}a^{8}-\frac{2}{7}a^{7}-\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{3}{7}a^{4}+\frac{3}{7}a^{3}+\frac{3}{11}a$, $\frac{1}{77}a^{12}+\frac{1}{7}a^{8}-\frac{3}{7}a^{7}-\frac{1}{7}a^{6}+\frac{2}{7}a^{5}+\frac{2}{7}a^{4}-\frac{3}{7}a^{3}+\frac{3}{11}a^{2}$, $\frac{1}{77}a^{13}-\frac{2}{7}a^{8}+\frac{3}{7}a^{7}+\frac{2}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}+\frac{10}{77}a^{3}$, $\frac{1}{77}a^{14}+\frac{1}{7}a^{8}+\frac{1}{7}a^{7}-\frac{3}{7}a^{6}+\frac{2}{7}a^{5}-\frac{34}{77}a^{4}+\frac{2}{7}a^{3}$, $\frac{1}{77}a^{15}+\frac{2}{7}a^{8}+\frac{1}{7}a^{7}+\frac{2}{7}a^{6}-\frac{12}{77}a^{5}-\frac{3}{7}a^{4}-\frac{1}{7}a^{3}$, $\frac{1}{145211747527}a^{16}-\frac{719298430}{145211747527}a^{15}-\frac{49280791}{20744535361}a^{14}+\frac{560965711}{145211747527}a^{13}+\frac{369482632}{145211747527}a^{12}+\frac{355272858}{145211747527}a^{11}-\frac{551907275}{145211747527}a^{10}-\frac{10336064685}{145211747527}a^{9}+\frac{29134572261}{145211747527}a^{8}-\frac{58615526671}{145211747527}a^{7}-\frac{6165988797}{20744535361}a^{6}+\frac{3767822654}{13201067957}a^{5}-\frac{5146997145}{20744535361}a^{4}-\frac{4542520419}{20744535361}a^{3}-\frac{6482095366}{20744535361}a^{2}-\frac{462637952}{1885866851}a+\frac{2916214}{171442441}$, $\frac{1}{34\!\cdots\!07}a^{17}+\frac{43\!\cdots\!08}{34\!\cdots\!07}a^{16}+\frac{20\!\cdots\!81}{49\!\cdots\!01}a^{15}-\frac{18\!\cdots\!14}{49\!\cdots\!01}a^{14}+\frac{82\!\cdots\!84}{34\!\cdots\!07}a^{13}+\frac{17\!\cdots\!82}{34\!\cdots\!07}a^{12}-\frac{55\!\cdots\!07}{34\!\cdots\!07}a^{11}+\frac{16\!\cdots\!64}{34\!\cdots\!07}a^{10}-\frac{43\!\cdots\!56}{34\!\cdots\!07}a^{9}-\frac{51\!\cdots\!44}{34\!\cdots\!07}a^{8}+\frac{18\!\cdots\!23}{34\!\cdots\!07}a^{7}-\frac{15\!\cdots\!84}{34\!\cdots\!07}a^{6}+\frac{63\!\cdots\!70}{34\!\cdots\!07}a^{5}+\frac{27\!\cdots\!02}{34\!\cdots\!07}a^{4}+\frac{17\!\cdots\!39}{34\!\cdots\!07}a^{3}-\frac{18\!\cdots\!22}{49\!\cdots\!01}a^{2}-\frac{36\!\cdots\!78}{45\!\cdots\!91}a-\frac{23\!\cdots\!75}{41\!\cdots\!81}$ 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:  $11$

Class group and class number

$C_{2}\times C_{2}\times C_{228}\times C_{74556}$, which has order $67995072$ (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:   $\frac{13\!\cdots\!66}{33\!\cdots\!59}a^{17}-\frac{10\!\cdots\!80}{33\!\cdots\!59}a^{16}+\frac{13\!\cdots\!96}{33\!\cdots\!59}a^{15}-\frac{91\!\cdots\!12}{30\!\cdots\!69}a^{14}+\frac{11\!\cdots\!08}{33\!\cdots\!59}a^{13}-\frac{68\!\cdots\!45}{33\!\cdots\!59}a^{12}+\frac{61\!\cdots\!00}{33\!\cdots\!59}a^{11}-\frac{32\!\cdots\!80}{33\!\cdots\!59}a^{10}+\frac{24\!\cdots\!72}{33\!\cdots\!59}a^{9}-\frac{12\!\cdots\!38}{33\!\cdots\!59}a^{8}+\frac{71\!\cdots\!22}{33\!\cdots\!59}a^{7}-\frac{37\!\cdots\!23}{33\!\cdots\!59}a^{6}+\frac{14\!\cdots\!44}{33\!\cdots\!59}a^{5}-\frac{81\!\cdots\!01}{33\!\cdots\!59}a^{4}+\frac{19\!\cdots\!06}{33\!\cdots\!59}a^{3}-\frac{11\!\cdots\!46}{33\!\cdots\!59}a^{2}+\frac{13\!\cdots\!44}{30\!\cdots\!69}a-\frac{74\!\cdots\!98}{27\!\cdots\!79}$, $\frac{25\!\cdots\!34}{33\!\cdots\!59}a^{17}-\frac{17\!\cdots\!48}{33\!\cdots\!59}a^{16}-\frac{11\!\cdots\!44}{33\!\cdots\!59}a^{15}-\frac{25\!\cdots\!99}{30\!\cdots\!69}a^{14}-\frac{23\!\cdots\!40}{33\!\cdots\!59}a^{13}-\frac{30\!\cdots\!64}{33\!\cdots\!59}a^{12}-\frac{27\!\cdots\!30}{33\!\cdots\!59}a^{11}-\frac{52\!\cdots\!52}{33\!\cdots\!59}a^{10}-\frac{17\!\cdots\!62}{33\!\cdots\!59}a^{9}+\frac{84\!\cdots\!26}{33\!\cdots\!59}a^{8}-\frac{57\!\cdots\!14}{33\!\cdots\!59}a^{7}+\frac{95\!\cdots\!67}{33\!\cdots\!59}a^{6}-\frac{12\!\cdots\!96}{33\!\cdots\!59}a^{5}+\frac{56\!\cdots\!69}{33\!\cdots\!59}a^{4}-\frac{11\!\cdots\!18}{33\!\cdots\!59}a^{3}+\frac{10\!\cdots\!50}{33\!\cdots\!59}a^{2}+\frac{17\!\cdots\!34}{30\!\cdots\!69}a+\frac{66\!\cdots\!51}{27\!\cdots\!79}$, $\frac{98\!\cdots\!98}{34\!\cdots\!07}a^{17}+\frac{39\!\cdots\!05}{34\!\cdots\!07}a^{16}+\frac{61\!\cdots\!28}{34\!\cdots\!07}a^{15}+\frac{28\!\cdots\!70}{34\!\cdots\!07}a^{14}-\frac{15\!\cdots\!24}{49\!\cdots\!01}a^{13}+\frac{23\!\cdots\!71}{34\!\cdots\!07}a^{12}+\frac{40\!\cdots\!44}{34\!\cdots\!07}a^{11}+\frac{12\!\cdots\!52}{34\!\cdots\!07}a^{10}+\frac{14\!\cdots\!52}{49\!\cdots\!01}a^{9}+\frac{53\!\cdots\!32}{34\!\cdots\!07}a^{8}+\frac{51\!\cdots\!38}{34\!\cdots\!07}a^{7}+\frac{16\!\cdots\!09}{34\!\cdots\!07}a^{6}+\frac{14\!\cdots\!44}{34\!\cdots\!07}a^{5}+\frac{34\!\cdots\!64}{34\!\cdots\!07}a^{4}+\frac{31\!\cdots\!14}{49\!\cdots\!01}a^{3}+\frac{69\!\cdots\!12}{49\!\cdots\!01}a^{2}+\frac{15\!\cdots\!56}{45\!\cdots\!91}a+\frac{41\!\cdots\!22}{41\!\cdots\!81}$, $\frac{63\!\cdots\!36}{34\!\cdots\!07}a^{17}-\frac{25\!\cdots\!35}{34\!\cdots\!07}a^{16}+\frac{57\!\cdots\!88}{34\!\cdots\!07}a^{15}-\frac{23\!\cdots\!70}{34\!\cdots\!07}a^{14}+\frac{63\!\cdots\!90}{49\!\cdots\!01}a^{13}-\frac{14\!\cdots\!66}{34\!\cdots\!07}a^{12}+\frac{24\!\cdots\!46}{34\!\cdots\!07}a^{11}-\frac{55\!\cdots\!70}{34\!\cdots\!07}a^{10}+\frac{98\!\cdots\!72}{34\!\cdots\!07}a^{9}-\frac{18\!\cdots\!74}{34\!\cdots\!07}a^{8}+\frac{30\!\cdots\!80}{34\!\cdots\!07}a^{7}-\frac{50\!\cdots\!73}{34\!\cdots\!07}a^{6}+\frac{62\!\cdots\!82}{34\!\cdots\!07}a^{5}-\frac{11\!\cdots\!45}{34\!\cdots\!07}a^{4}+\frac{12\!\cdots\!32}{49\!\cdots\!01}a^{3}-\frac{25\!\cdots\!79}{49\!\cdots\!01}a^{2}+\frac{87\!\cdots\!92}{45\!\cdots\!91}a-\frac{15\!\cdots\!17}{41\!\cdots\!81}$, $\frac{18\!\cdots\!36}{34\!\cdots\!07}a^{17}+\frac{29\!\cdots\!88}{34\!\cdots\!07}a^{16}+\frac{10\!\cdots\!44}{34\!\cdots\!07}a^{15}-\frac{14\!\cdots\!69}{34\!\cdots\!07}a^{14}+\frac{78\!\cdots\!62}{34\!\cdots\!07}a^{13}+\frac{14\!\cdots\!18}{34\!\cdots\!07}a^{12}+\frac{43\!\cdots\!68}{34\!\cdots\!07}a^{11}+\frac{14\!\cdots\!00}{49\!\cdots\!01}a^{10}+\frac{16\!\cdots\!74}{34\!\cdots\!07}a^{9}+\frac{32\!\cdots\!55}{34\!\cdots\!07}a^{8}+\frac{40\!\cdots\!56}{34\!\cdots\!07}a^{7}+\frac{44\!\cdots\!39}{34\!\cdots\!07}a^{6}+\frac{58\!\cdots\!06}{34\!\cdots\!07}a^{5}-\frac{53\!\cdots\!35}{34\!\cdots\!07}a^{4}+\frac{43\!\cdots\!76}{49\!\cdots\!01}a^{3}-\frac{34\!\cdots\!41}{49\!\cdots\!01}a^{2}-\frac{15\!\cdots\!10}{45\!\cdots\!91}a-\frac{21\!\cdots\!63}{41\!\cdots\!81}$, $\frac{13\!\cdots\!52}{31\!\cdots\!37}a^{17}-\frac{59\!\cdots\!04}{31\!\cdots\!37}a^{16}+\frac{12\!\cdots\!14}{45\!\cdots\!91}a^{15}-\frac{47\!\cdots\!16}{45\!\cdots\!91}a^{14}+\frac{60\!\cdots\!00}{31\!\cdots\!37}a^{13}-\frac{18\!\cdots\!72}{31\!\cdots\!37}a^{12}+\frac{28\!\cdots\!80}{31\!\cdots\!37}a^{11}-\frac{47\!\cdots\!91}{31\!\cdots\!37}a^{10}+\frac{91\!\cdots\!14}{31\!\cdots\!37}a^{9}-\frac{24\!\cdots\!46}{45\!\cdots\!91}a^{8}+\frac{20\!\cdots\!58}{31\!\cdots\!37}a^{7}-\frac{54\!\cdots\!64}{28\!\cdots\!67}a^{6}+\frac{20\!\cdots\!00}{31\!\cdots\!37}a^{5}-\frac{16\!\cdots\!48}{31\!\cdots\!37}a^{4}-\frac{58\!\cdots\!82}{45\!\cdots\!91}a^{3}-\frac{43\!\cdots\!00}{41\!\cdots\!81}a^{2}-\frac{14\!\cdots\!88}{41\!\cdots\!81}a-\frac{43\!\cdots\!61}{37\!\cdots\!71}$, $\frac{27\!\cdots\!74}{49\!\cdots\!01}a^{17}+\frac{86\!\cdots\!58}{34\!\cdots\!07}a^{16}+\frac{60\!\cdots\!14}{34\!\cdots\!07}a^{15}+\frac{72\!\cdots\!88}{34\!\cdots\!07}a^{14}+\frac{23\!\cdots\!18}{34\!\cdots\!07}a^{13}+\frac{63\!\cdots\!18}{34\!\cdots\!07}a^{12}+\frac{22\!\cdots\!26}{34\!\cdots\!07}a^{11}+\frac{43\!\cdots\!57}{34\!\cdots\!07}a^{10}-\frac{33\!\cdots\!46}{34\!\cdots\!07}a^{9}+\frac{18\!\cdots\!54}{34\!\cdots\!07}a^{8}-\frac{18\!\cdots\!08}{34\!\cdots\!07}a^{7}+\frac{56\!\cdots\!18}{34\!\cdots\!07}a^{6}-\frac{74\!\cdots\!54}{34\!\cdots\!07}a^{5}+\frac{17\!\cdots\!37}{49\!\cdots\!01}a^{4}-\frac{23\!\cdots\!52}{49\!\cdots\!01}a^{3}+\frac{21\!\cdots\!17}{49\!\cdots\!01}a^{2}-\frac{17\!\cdots\!12}{45\!\cdots\!91}a+\frac{89\!\cdots\!72}{41\!\cdots\!81}$, $\frac{17\!\cdots\!72}{34\!\cdots\!07}a^{17}+\frac{67\!\cdots\!33}{34\!\cdots\!07}a^{16}+\frac{78\!\cdots\!90}{34\!\cdots\!07}a^{15}+\frac{80\!\cdots\!67}{34\!\cdots\!07}a^{14}+\frac{66\!\cdots\!82}{49\!\cdots\!01}a^{13}+\frac{70\!\cdots\!41}{34\!\cdots\!07}a^{12}+\frac{34\!\cdots\!72}{49\!\cdots\!01}a^{11}+\frac{43\!\cdots\!11}{34\!\cdots\!07}a^{10}+\frac{10\!\cdots\!28}{34\!\cdots\!07}a^{9}+\frac{18\!\cdots\!83}{34\!\cdots\!07}a^{8}+\frac{34\!\cdots\!74}{34\!\cdots\!07}a^{7}+\frac{54\!\cdots\!23}{34\!\cdots\!07}a^{6}+\frac{73\!\cdots\!98}{34\!\cdots\!07}a^{5}+\frac{10\!\cdots\!82}{34\!\cdots\!07}a^{4}+\frac{14\!\cdots\!02}{49\!\cdots\!01}a^{3}+\frac{18\!\cdots\!53}{49\!\cdots\!01}a^{2}+\frac{10\!\cdots\!38}{45\!\cdots\!91}a+\frac{81\!\cdots\!27}{41\!\cdots\!81}$ 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:  \( 7595459.562747852 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 7595459.562747852 \cdot 67995072}{2\cdot\sqrt{7065768916593110856047783889584316216508416}}\cr\approx \mathstrut & 1.48265667982827 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 98*x^16 - 230*x^15 + 7698*x^14 - 12222*x^13 + 450829*x^12 - 294056*x^11 + 20316110*x^10 - 1036720*x^9 + 691813801*x^8 + 163570074*x^7 + 17189420781*x^6 + 4958427054*x^5 + 301709383024*x^4 + 51432868270*x^3 + 3384649707670*x^2 + 106026030264*x + 17934199170349)
 
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 - 2*x^17 + 98*x^16 - 230*x^15 + 7698*x^14 - 12222*x^13 + 450829*x^12 - 294056*x^11 + 20316110*x^10 - 1036720*x^9 + 691813801*x^8 + 163570074*x^7 + 17189420781*x^6 + 4958427054*x^5 + 301709383024*x^4 + 51432868270*x^3 + 3384649707670*x^2 + 106026030264*x + 17934199170349, 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 - 2*x^17 + 98*x^16 - 230*x^15 + 7698*x^14 - 12222*x^13 + 450829*x^12 - 294056*x^11 + 20316110*x^10 - 1036720*x^9 + 691813801*x^8 + 163570074*x^7 + 17189420781*x^6 + 4958427054*x^5 + 301709383024*x^4 + 51432868270*x^3 + 3384649707670*x^2 + 106026030264*x + 17934199170349);
 
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 - 2*x^17 + 98*x^16 - 230*x^15 + 7698*x^14 - 12222*x^13 + 450829*x^12 - 294056*x^11 + 20316110*x^10 - 1036720*x^9 + 691813801*x^8 + 163570074*x^7 + 17189420781*x^6 + 4958427054*x^5 + 301709383024*x^4 + 51432868270*x^3 + 3384649707670*x^2 + 106026030264*x + 17934199170349);
 
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

$C_{18}$ (as 18T1):

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 cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-21}) \), 3.3.361.1, 6.0.77241777984.6, 9.9.1998099208210609.1

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]
 

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 ${\href{/padicField/5.9.0.1}{9} }^{2}$ R ${\href{/padicField/11.1.0.1}{1} }^{18}$ $18$ ${\href{/padicField/17.9.0.1}{9} }^{2}$ R ${\href{/padicField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ $18$ $18$ $18$ $18$

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 2.18.18.118$x^{18} + 18 x^{17} + 198 x^{16} + 1536 x^{15} + 9312 x^{14} + 45696 x^{13} + 187776 x^{12} + 655872 x^{11} + 2010400 x^{10} + 5500224 x^{9} + 14116288 x^{8} + 34058240 x^{7} + 79898624 x^{6} + 169960448 x^{5} + 335809536 x^{4} + 542121984 x^{3} + 798549248 x^{2} + 783239680 x + 807955968$$2$$9$$18$$C_{18}$$[2]^{9}$
\(3\) Copy content Toggle raw display Deg $18$$2$$9$$9$
\(7\) Copy content Toggle raw display 7.6.5.1$x^{6} + 21$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.1$x^{6} + 21$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.1$x^{6} + 21$$6$$1$$5$$C_6$$[\ ]_{6}$
\(19\) Copy content Toggle raw display 19.9.8.3$x^{9} + 152$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.3$x^{9} + 152$$9$$1$$8$$C_9$$[\ ]_{9}$