Properties

Label 18.6.146...693.1
Degree $18$
Signature $[6, 6]$
Discriminant $1.461\times 10^{25}$
Root discriminant \(25.01\)
Ramified primes $19,37$
Class number $1$
Class group trivial
Galois group $C_2^2:C_{18}$ (as 18T26)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 4*x^16 + 33*x^15 - 73*x^14 + 6*x^13 + 269*x^12 - 365*x^11 + 9*x^10 + 331*x^9 + 106*x^8 - 1784*x^7 + 1952*x^6 + 505*x^5 - 976*x^4 - 985*x^3 + 2554*x^2 - 1500*x - 113)
 
gp: K = bnfinit(y^18 - 3*y^17 - 4*y^16 + 33*y^15 - 73*y^14 + 6*y^13 + 269*y^12 - 365*y^11 + 9*y^10 + 331*y^9 + 106*y^8 - 1784*y^7 + 1952*y^6 + 505*y^5 - 976*y^4 - 985*y^3 + 2554*y^2 - 1500*y - 113, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 4*x^16 + 33*x^15 - 73*x^14 + 6*x^13 + 269*x^12 - 365*x^11 + 9*x^10 + 331*x^9 + 106*x^8 - 1784*x^7 + 1952*x^6 + 505*x^5 - 976*x^4 - 985*x^3 + 2554*x^2 - 1500*x - 113);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 4*x^16 + 33*x^15 - 73*x^14 + 6*x^13 + 269*x^12 - 365*x^11 + 9*x^10 + 331*x^9 + 106*x^8 - 1784*x^7 + 1952*x^6 + 505*x^5 - 976*x^4 - 985*x^3 + 2554*x^2 - 1500*x - 113)
 

\( x^{18} - 3 x^{17} - 4 x^{16} + 33 x^{15} - 73 x^{14} + 6 x^{13} + 269 x^{12} - 365 x^{11} + 9 x^{10} + 331 x^{9} + 106 x^{8} - 1784 x^{7} + 1952 x^{6} + 505 x^{5} - 976 x^{4} - 985 x^{3} + \cdots - 113 \) 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:  $[6, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(14610422921440715006545693\) \(\medspace = 19^{16}\cdot 37^{3}\) 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:  \(25.01\)
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:  $19^{8/9}37^{1/2}\approx 83.32411881951172$
Ramified primes:   \(19\), \(37\) 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:  \(\Q(\sqrt{37}) \)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7}a^{15}-\frac{1}{7}a^{14}-\frac{1}{7}a^{13}+\frac{1}{7}a^{12}-\frac{2}{7}a^{11}-\frac{3}{7}a^{10}-\frac{3}{7}a^{9}-\frac{1}{7}a^{7}-\frac{2}{7}a^{6}-\frac{1}{7}a^{5}+\frac{2}{7}a^{3}+\frac{2}{7}a^{2}-\frac{3}{7}a-\frac{2}{7}$, $\frac{1}{259}a^{16}+\frac{2}{37}a^{15}-\frac{79}{259}a^{14}-\frac{15}{37}a^{13}-\frac{71}{259}a^{12}-\frac{82}{259}a^{11}+\frac{8}{259}a^{10}+\frac{18}{259}a^{9}-\frac{43}{259}a^{8}-\frac{3}{259}a^{7}+\frac{46}{259}a^{6}+\frac{48}{259}a^{5}-\frac{12}{259}a^{4}-\frac{108}{259}a^{3}-\frac{92}{259}a^{2}-\frac{47}{259}a+\frac{26}{259}$, $\frac{1}{12\!\cdots\!93}a^{17}-\frac{12\!\cdots\!25}{12\!\cdots\!93}a^{16}-\frac{10\!\cdots\!95}{12\!\cdots\!93}a^{15}-\frac{24\!\cdots\!92}{12\!\cdots\!93}a^{14}+\frac{22\!\cdots\!72}{12\!\cdots\!93}a^{13}-\frac{47\!\cdots\!43}{12\!\cdots\!93}a^{12}-\frac{70\!\cdots\!62}{18\!\cdots\!99}a^{11}-\frac{43\!\cdots\!07}{12\!\cdots\!93}a^{10}-\frac{60\!\cdots\!34}{18\!\cdots\!99}a^{9}+\frac{20\!\cdots\!04}{12\!\cdots\!93}a^{8}-\frac{62\!\cdots\!58}{12\!\cdots\!93}a^{7}+\frac{56\!\cdots\!06}{12\!\cdots\!93}a^{6}-\frac{67\!\cdots\!25}{18\!\cdots\!99}a^{5}+\frac{15\!\cdots\!78}{12\!\cdots\!93}a^{4}-\frac{32\!\cdots\!36}{12\!\cdots\!93}a^{3}+\frac{60\!\cdots\!05}{12\!\cdots\!93}a^{2}+\frac{43\!\cdots\!99}{12\!\cdots\!93}a+\frac{19\!\cdots\!77}{12\!\cdots\!93}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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:  $11$
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:   $\frac{84\!\cdots\!20}{12\!\cdots\!93}a^{17}-\frac{20\!\cdots\!52}{12\!\cdots\!93}a^{16}-\frac{18\!\cdots\!71}{12\!\cdots\!93}a^{15}+\frac{18\!\cdots\!70}{12\!\cdots\!93}a^{14}-\frac{58\!\cdots\!56}{12\!\cdots\!93}a^{13}+\frac{60\!\cdots\!77}{12\!\cdots\!93}a^{12}+\frac{32\!\cdots\!81}{12\!\cdots\!93}a^{11}-\frac{25\!\cdots\!23}{18\!\cdots\!99}a^{10}+\frac{54\!\cdots\!59}{12\!\cdots\!93}a^{9}-\frac{73\!\cdots\!93}{12\!\cdots\!93}a^{8}+\frac{59\!\cdots\!04}{12\!\cdots\!93}a^{7}-\frac{80\!\cdots\!11}{12\!\cdots\!93}a^{6}+\frac{85\!\cdots\!16}{12\!\cdots\!93}a^{5}-\frac{18\!\cdots\!89}{12\!\cdots\!93}a^{4}+\frac{26\!\cdots\!14}{12\!\cdots\!93}a^{3}-\frac{18\!\cdots\!65}{12\!\cdots\!93}a^{2}-\frac{10\!\cdots\!63}{12\!\cdots\!93}a+\frac{86\!\cdots\!65}{12\!\cdots\!93}$, $\frac{21\!\cdots\!70}{12\!\cdots\!93}a^{17}-\frac{57\!\cdots\!44}{12\!\cdots\!93}a^{16}-\frac{11\!\cdots\!24}{12\!\cdots\!93}a^{15}+\frac{68\!\cdots\!96}{12\!\cdots\!93}a^{14}-\frac{13\!\cdots\!88}{12\!\cdots\!93}a^{13}-\frac{40\!\cdots\!12}{12\!\cdots\!93}a^{12}+\frac{58\!\cdots\!04}{12\!\cdots\!93}a^{11}-\frac{56\!\cdots\!17}{12\!\cdots\!93}a^{10}-\frac{21\!\cdots\!02}{12\!\cdots\!93}a^{9}+\frac{68\!\cdots\!05}{12\!\cdots\!93}a^{8}+\frac{34\!\cdots\!97}{12\!\cdots\!93}a^{7}-\frac{38\!\cdots\!46}{12\!\cdots\!93}a^{6}+\frac{28\!\cdots\!24}{12\!\cdots\!93}a^{5}+\frac{22\!\cdots\!71}{12\!\cdots\!93}a^{4}-\frac{15\!\cdots\!62}{12\!\cdots\!93}a^{3}-\frac{23\!\cdots\!45}{12\!\cdots\!93}a^{2}+\frac{56\!\cdots\!01}{12\!\cdots\!93}a-\frac{20\!\cdots\!19}{12\!\cdots\!93}$, $\frac{61\!\cdots\!74}{12\!\cdots\!93}a^{17}+\frac{12\!\cdots\!30}{12\!\cdots\!93}a^{16}-\frac{94\!\cdots\!33}{12\!\cdots\!93}a^{15}+\frac{68\!\cdots\!72}{12\!\cdots\!93}a^{14}+\frac{39\!\cdots\!81}{12\!\cdots\!93}a^{13}-\frac{18\!\cdots\!52}{12\!\cdots\!93}a^{12}+\frac{15\!\cdots\!60}{12\!\cdots\!93}a^{11}+\frac{38\!\cdots\!43}{12\!\cdots\!93}a^{10}-\frac{79\!\cdots\!46}{12\!\cdots\!93}a^{9}+\frac{58\!\cdots\!99}{12\!\cdots\!93}a^{8}+\frac{32\!\cdots\!07}{12\!\cdots\!93}a^{7}-\frac{43\!\cdots\!86}{12\!\cdots\!93}a^{6}-\frac{25\!\cdots\!22}{12\!\cdots\!93}a^{5}+\frac{35\!\cdots\!32}{12\!\cdots\!93}a^{4}-\frac{49\!\cdots\!60}{12\!\cdots\!93}a^{3}-\frac{10\!\cdots\!88}{12\!\cdots\!93}a^{2}+\frac{11\!\cdots\!83}{12\!\cdots\!93}a+\frac{86\!\cdots\!77}{12\!\cdots\!93}$, $\frac{14\!\cdots\!85}{12\!\cdots\!93}a^{17}-\frac{25\!\cdots\!63}{18\!\cdots\!99}a^{16}+\frac{31\!\cdots\!37}{12\!\cdots\!93}a^{15}+\frac{97\!\cdots\!94}{12\!\cdots\!93}a^{14}-\frac{49\!\cdots\!83}{12\!\cdots\!93}a^{13}+\frac{97\!\cdots\!26}{12\!\cdots\!93}a^{12}+\frac{79\!\cdots\!71}{12\!\cdots\!93}a^{11}-\frac{33\!\cdots\!19}{12\!\cdots\!93}a^{10}+\frac{44\!\cdots\!99}{12\!\cdots\!93}a^{9}-\frac{30\!\cdots\!57}{12\!\cdots\!93}a^{8}-\frac{42\!\cdots\!93}{12\!\cdots\!93}a^{7}-\frac{43\!\cdots\!37}{12\!\cdots\!93}a^{6}+\frac{21\!\cdots\!45}{12\!\cdots\!93}a^{5}-\frac{14\!\cdots\!82}{12\!\cdots\!93}a^{4}+\frac{43\!\cdots\!16}{12\!\cdots\!93}a^{3}+\frac{19\!\cdots\!07}{12\!\cdots\!93}a^{2}+\frac{55\!\cdots\!28}{12\!\cdots\!93}a-\frac{24\!\cdots\!86}{12\!\cdots\!93}$, $\frac{11\!\cdots\!82}{12\!\cdots\!93}a^{17}-\frac{13\!\cdots\!17}{18\!\cdots\!99}a^{16}-\frac{10\!\cdots\!10}{12\!\cdots\!93}a^{15}+\frac{22\!\cdots\!01}{12\!\cdots\!93}a^{14}-\frac{10\!\cdots\!32}{12\!\cdots\!93}a^{13}-\frac{10\!\cdots\!55}{12\!\cdots\!93}a^{12}+\frac{19\!\cdots\!21}{12\!\cdots\!93}a^{11}+\frac{21\!\cdots\!35}{12\!\cdots\!93}a^{10}-\frac{28\!\cdots\!98}{12\!\cdots\!93}a^{9}-\frac{22\!\cdots\!78}{12\!\cdots\!93}a^{8}+\frac{36\!\cdots\!12}{12\!\cdots\!93}a^{7}-\frac{97\!\cdots\!06}{12\!\cdots\!93}a^{6}-\frac{12\!\cdots\!49}{12\!\cdots\!93}a^{5}+\frac{16\!\cdots\!90}{12\!\cdots\!93}a^{4}+\frac{34\!\cdots\!64}{12\!\cdots\!93}a^{3}+\frac{69\!\cdots\!05}{12\!\cdots\!93}a^{2}-\frac{12\!\cdots\!73}{12\!\cdots\!93}a-\frac{18\!\cdots\!63}{12\!\cdots\!93}$, $\frac{70\!\cdots\!62}{12\!\cdots\!93}a^{17}-\frac{32\!\cdots\!31}{12\!\cdots\!93}a^{16}-\frac{86\!\cdots\!20}{12\!\cdots\!93}a^{15}+\frac{24\!\cdots\!85}{12\!\cdots\!93}a^{14}-\frac{79\!\cdots\!64}{12\!\cdots\!93}a^{13}+\frac{13\!\cdots\!24}{18\!\cdots\!99}a^{12}+\frac{11\!\cdots\!18}{12\!\cdots\!93}a^{11}-\frac{36\!\cdots\!95}{12\!\cdots\!93}a^{10}+\frac{70\!\cdots\!86}{18\!\cdots\!99}a^{9}-\frac{42\!\cdots\!41}{12\!\cdots\!93}a^{8}+\frac{42\!\cdots\!09}{12\!\cdots\!93}a^{7}-\frac{20\!\cdots\!30}{18\!\cdots\!99}a^{6}+\frac{24\!\cdots\!69}{18\!\cdots\!99}a^{5}-\frac{15\!\cdots\!60}{12\!\cdots\!93}a^{4}+\frac{14\!\cdots\!95}{18\!\cdots\!99}a^{3}-\frac{13\!\cdots\!42}{18\!\cdots\!99}a^{2}+\frac{75\!\cdots\!32}{12\!\cdots\!93}a+\frac{65\!\cdots\!74}{12\!\cdots\!93}$, $\frac{29\!\cdots\!17}{12\!\cdots\!93}a^{17}-\frac{61\!\cdots\!72}{12\!\cdots\!93}a^{16}-\frac{17\!\cdots\!23}{12\!\cdots\!93}a^{15}+\frac{78\!\cdots\!05}{12\!\cdots\!93}a^{14}-\frac{13\!\cdots\!85}{12\!\cdots\!93}a^{13}-\frac{13\!\cdots\!28}{18\!\cdots\!99}a^{12}+\frac{65\!\cdots\!63}{12\!\cdots\!93}a^{11}-\frac{41\!\cdots\!43}{12\!\cdots\!93}a^{10}-\frac{33\!\cdots\!80}{18\!\cdots\!99}a^{9}+\frac{38\!\cdots\!87}{12\!\cdots\!93}a^{8}+\frac{83\!\cdots\!97}{12\!\cdots\!93}a^{7}-\frac{60\!\cdots\!58}{18\!\cdots\!99}a^{6}+\frac{21\!\cdots\!07}{18\!\cdots\!99}a^{5}+\frac{20\!\cdots\!84}{12\!\cdots\!93}a^{4}+\frac{18\!\cdots\!30}{18\!\cdots\!99}a^{3}-\frac{27\!\cdots\!06}{18\!\cdots\!99}a^{2}+\frac{31\!\cdots\!74}{12\!\cdots\!93}a-\frac{39\!\cdots\!05}{12\!\cdots\!93}$, $\frac{18\!\cdots\!70}{12\!\cdots\!93}a^{17}-\frac{25\!\cdots\!26}{12\!\cdots\!93}a^{16}-\frac{12\!\cdots\!59}{12\!\cdots\!93}a^{15}+\frac{44\!\cdots\!68}{12\!\cdots\!93}a^{14}-\frac{58\!\cdots\!88}{12\!\cdots\!93}a^{13}-\frac{12\!\cdots\!92}{12\!\cdots\!93}a^{12}+\frac{40\!\cdots\!01}{12\!\cdots\!93}a^{11}-\frac{89\!\cdots\!90}{12\!\cdots\!93}a^{10}-\frac{39\!\cdots\!16}{12\!\cdots\!93}a^{9}+\frac{51\!\cdots\!16}{12\!\cdots\!93}a^{8}+\frac{36\!\cdots\!86}{12\!\cdots\!93}a^{7}-\frac{22\!\cdots\!69}{12\!\cdots\!93}a^{6}+\frac{14\!\cdots\!44}{12\!\cdots\!93}a^{5}+\frac{19\!\cdots\!95}{12\!\cdots\!93}a^{4}+\frac{18\!\cdots\!10}{12\!\cdots\!93}a^{3}-\frac{90\!\cdots\!33}{12\!\cdots\!93}a^{2}+\frac{16\!\cdots\!00}{12\!\cdots\!93}a-\frac{90\!\cdots\!13}{12\!\cdots\!93}$, $\frac{98\!\cdots\!89}{12\!\cdots\!93}a^{17}-\frac{19\!\cdots\!15}{12\!\cdots\!93}a^{16}-\frac{64\!\cdots\!47}{12\!\cdots\!93}a^{15}+\frac{13\!\cdots\!20}{12\!\cdots\!93}a^{14}-\frac{15\!\cdots\!89}{12\!\cdots\!93}a^{13}-\frac{10\!\cdots\!67}{18\!\cdots\!99}a^{12}+\frac{87\!\cdots\!76}{12\!\cdots\!93}a^{11}+\frac{49\!\cdots\!66}{12\!\cdots\!93}a^{10}+\frac{43\!\cdots\!17}{18\!\cdots\!99}a^{9}+\frac{10\!\cdots\!64}{12\!\cdots\!93}a^{8}+\frac{34\!\cdots\!54}{12\!\cdots\!93}a^{7}-\frac{53\!\cdots\!80}{18\!\cdots\!99}a^{6}-\frac{12\!\cdots\!06}{18\!\cdots\!99}a^{5}+\frac{48\!\cdots\!16}{12\!\cdots\!93}a^{4}+\frac{21\!\cdots\!47}{18\!\cdots\!99}a^{3}+\frac{20\!\cdots\!02}{18\!\cdots\!99}a^{2}-\frac{18\!\cdots\!66}{12\!\cdots\!93}a+\frac{40\!\cdots\!07}{12\!\cdots\!93}$, $\frac{49\!\cdots\!67}{12\!\cdots\!93}a^{17}-\frac{12\!\cdots\!26}{12\!\cdots\!93}a^{16}-\frac{22\!\cdots\!14}{12\!\cdots\!93}a^{15}+\frac{14\!\cdots\!89}{12\!\cdots\!93}a^{14}-\frac{31\!\cdots\!71}{12\!\cdots\!93}a^{13}-\frac{52\!\cdots\!66}{18\!\cdots\!99}a^{12}+\frac{11\!\cdots\!85}{12\!\cdots\!93}a^{11}-\frac{14\!\cdots\!77}{12\!\cdots\!93}a^{10}-\frac{32\!\cdots\!93}{18\!\cdots\!99}a^{9}+\frac{13\!\cdots\!53}{12\!\cdots\!93}a^{8}+\frac{90\!\cdots\!87}{12\!\cdots\!93}a^{7}-\frac{11\!\cdots\!57}{18\!\cdots\!99}a^{6}+\frac{10\!\cdots\!53}{18\!\cdots\!99}a^{5}+\frac{21\!\cdots\!50}{12\!\cdots\!93}a^{4}-\frac{48\!\cdots\!75}{18\!\cdots\!99}a^{3}-\frac{78\!\cdots\!43}{18\!\cdots\!99}a^{2}+\frac{10\!\cdots\!63}{12\!\cdots\!93}a-\frac{14\!\cdots\!87}{34\!\cdots\!89}$, $\frac{12\!\cdots\!03}{12\!\cdots\!93}a^{17}-\frac{19\!\cdots\!78}{12\!\cdots\!93}a^{16}-\frac{71\!\cdots\!91}{12\!\cdots\!93}a^{15}+\frac{29\!\cdots\!62}{12\!\cdots\!93}a^{14}-\frac{52\!\cdots\!74}{12\!\cdots\!93}a^{13}-\frac{49\!\cdots\!99}{12\!\cdots\!93}a^{12}+\frac{23\!\cdots\!77}{12\!\cdots\!93}a^{11}-\frac{35\!\cdots\!43}{34\!\cdots\!89}a^{10}-\frac{71\!\cdots\!22}{12\!\cdots\!93}a^{9}+\frac{29\!\cdots\!70}{12\!\cdots\!93}a^{8}+\frac{49\!\cdots\!47}{12\!\cdots\!93}a^{7}-\frac{47\!\cdots\!59}{34\!\cdots\!89}a^{6}+\frac{40\!\cdots\!41}{12\!\cdots\!93}a^{5}+\frac{46\!\cdots\!03}{18\!\cdots\!99}a^{4}-\frac{26\!\cdots\!20}{12\!\cdots\!93}a^{3}-\frac{13\!\cdots\!67}{12\!\cdots\!93}a^{2}+\frac{19\!\cdots\!74}{12\!\cdots\!93}a+\frac{28\!\cdots\!81}{12\!\cdots\!93}$ Copy content Toggle raw display
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:  \( 290338.751062 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 290338.751062 \cdot 1}{2\cdot\sqrt{14610422921440715006545693}}\cr\approx \mathstrut & 0.149555641167 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 4*x^16 + 33*x^15 - 73*x^14 + 6*x^13 + 269*x^12 - 365*x^11 + 9*x^10 + 331*x^9 + 106*x^8 - 1784*x^7 + 1952*x^6 + 505*x^5 - 976*x^4 - 985*x^3 + 2554*x^2 - 1500*x - 113)
 
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^17 - 4*x^16 + 33*x^15 - 73*x^14 + 6*x^13 + 269*x^12 - 365*x^11 + 9*x^10 + 331*x^9 + 106*x^8 - 1784*x^7 + 1952*x^6 + 505*x^5 - 976*x^4 - 985*x^3 + 2554*x^2 - 1500*x - 113, 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^17 - 4*x^16 + 33*x^15 - 73*x^14 + 6*x^13 + 269*x^12 - 365*x^11 + 9*x^10 + 331*x^9 + 106*x^8 - 1784*x^7 + 1952*x^6 + 505*x^5 - 976*x^4 - 985*x^3 + 2554*x^2 - 1500*x - 113);
 
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^17 - 4*x^16 + 33*x^15 - 73*x^14 + 6*x^13 + 269*x^12 - 365*x^11 + 9*x^10 + 331*x^9 + 106*x^8 - 1784*x^7 + 1952*x^6 + 505*x^5 - 976*x^4 - 985*x^3 + 2554*x^2 - 1500*x - 113);
 
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

$C_2^2:C_{18}$ (as 18T26):

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 72
The 24 conjugacy class representatives for $C_2^2:C_{18}$
Character table for $C_2^2:C_{18}$ is not computed

Intermediate fields

3.3.361.1, 6.2.4821877.1, \(\Q(\zeta_{19})^+\)

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 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $18$ ${\href{/padicField/3.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{6}$ $18$ $18$ R $18$ $18$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ R ${\href{/padicField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/47.9.0.1}{9} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }^{2}$ $18$

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
\(19\) Copy content Toggle raw display 19.18.16.1$x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$$9$$2$$16$$C_{18}$$[\ ]_{9}^{2}$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$