Properties

Label 18.18.160...625.1
Degree $18$
Signature $[18, 0]$
Discriminant $1.610\times 10^{31}$
Root discriminant \(54.16\)
Ramified primes $3,5,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\He_3:C_4$ (as 18T49)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 18*x^16 + 141*x^15 + 108*x^14 - 1227*x^13 - 438*x^12 + 5220*x^11 + 1956*x^10 - 11198*x^9 - 5475*x^8 + 11283*x^7 + 6315*x^6 - 5184*x^5 - 3051*x^4 + 930*x^3 + 552*x^2 - 24*x - 16)
 
gp: K = bnfinit(y^18 - 6*y^17 - 18*y^16 + 141*y^15 + 108*y^14 - 1227*y^13 - 438*y^12 + 5220*y^11 + 1956*y^10 - 11198*y^9 - 5475*y^8 + 11283*y^7 + 6315*y^6 - 5184*y^5 - 3051*y^4 + 930*y^3 + 552*y^2 - 24*y - 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 - 18*x^16 + 141*x^15 + 108*x^14 - 1227*x^13 - 438*x^12 + 5220*x^11 + 1956*x^10 - 11198*x^9 - 5475*x^8 + 11283*x^7 + 6315*x^6 - 5184*x^5 - 3051*x^4 + 930*x^3 + 552*x^2 - 24*x - 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 - 18*x^16 + 141*x^15 + 108*x^14 - 1227*x^13 - 438*x^12 + 5220*x^11 + 1956*x^10 - 11198*x^9 - 5475*x^8 + 11283*x^7 + 6315*x^6 - 5184*x^5 - 3051*x^4 + 930*x^3 + 552*x^2 - 24*x - 16)
 

\( x^{18} - 6 x^{17} - 18 x^{16} + 141 x^{15} + 108 x^{14} - 1227 x^{13} - 438 x^{12} + 5220 x^{11} + 1956 x^{10} - 11198 x^{9} - 5475 x^{8} + 11283 x^{7} + 6315 x^{6} - 5184 x^{5} + \cdots - 16 \) 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:  $[18, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(16096057926792443193781494140625\) \(\medspace = 3^{26}\cdot 5^{12}\cdot 11^{10}\) 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:  \(54.16\)
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:  $3^{85/54}5^{3/4}11^{2/3}\approx 93.22094273535318$
Ramified primes:   \(3\), \(5\), \(11\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{37\!\cdots\!68}a^{17}+\frac{71\!\cdots\!83}{94\!\cdots\!92}a^{16}+\frac{10\!\cdots\!91}{18\!\cdots\!84}a^{15}+\frac{44\!\cdots\!45}{37\!\cdots\!68}a^{14}-\frac{16\!\cdots\!61}{18\!\cdots\!84}a^{13}-\frac{36\!\cdots\!91}{37\!\cdots\!68}a^{12}-\frac{23\!\cdots\!25}{94\!\cdots\!92}a^{11}-\frac{43\!\cdots\!81}{94\!\cdots\!92}a^{10}+\frac{31\!\cdots\!15}{94\!\cdots\!92}a^{9}+\frac{51\!\cdots\!85}{18\!\cdots\!84}a^{8}+\frac{40\!\cdots\!49}{37\!\cdots\!68}a^{7}+\frac{10\!\cdots\!09}{37\!\cdots\!68}a^{6}+\frac{16\!\cdots\!53}{37\!\cdots\!68}a^{5}-\frac{41\!\cdots\!39}{18\!\cdots\!84}a^{4}-\frac{39\!\cdots\!67}{37\!\cdots\!68}a^{3}+\frac{49\!\cdots\!17}{94\!\cdots\!92}a^{2}+\frac{96\!\cdots\!27}{23\!\cdots\!48}a-\frac{12\!\cdots\!51}{47\!\cdots\!96}$ 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

Trivial group, which has order $1$ (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:  $17$
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:   $\frac{210228674823}{4495183640512}a^{17}-\frac{333656980311}{1123795910128}a^{16}-\frac{1568915003739}{2247591820256}a^{15}+\frac{29630734176591}{4495183640512}a^{14}+\frac{4669835956473}{2247591820256}a^{13}-\frac{236079774883209}{4495183640512}a^{12}+\frac{1386225793605}{1123795910128}a^{11}+\frac{221926009061193}{1123795910128}a^{10}+\frac{12345341245681}{1123795910128}a^{9}-\frac{775022287054389}{2247591820256}a^{8}-\frac{349422813396393}{4495183640512}a^{7}+\frac{942031494560691}{4495183640512}a^{6}+\frac{261278007000147}{4495183640512}a^{5}-\frac{3750313132797}{2247591820256}a^{4}+\frac{22479559006527}{4495183640512}a^{3}-\frac{23630818341045}{1123795910128}a^{2}-\frac{1779153925239}{280948977532}a+\frac{126818574567}{561897955064}$, $\frac{17\!\cdots\!53}{37\!\cdots\!68}a^{17}-\frac{12\!\cdots\!49}{94\!\cdots\!92}a^{16}-\frac{32\!\cdots\!13}{18\!\cdots\!84}a^{15}+\frac{14\!\cdots\!13}{37\!\cdots\!68}a^{14}+\frac{47\!\cdots\!63}{18\!\cdots\!84}a^{13}-\frac{15\!\cdots\!19}{37\!\cdots\!68}a^{12}-\frac{17\!\cdots\!77}{94\!\cdots\!92}a^{11}+\frac{17\!\cdots\!91}{94\!\cdots\!92}a^{10}+\frac{70\!\cdots\!31}{94\!\cdots\!92}a^{9}-\frac{48\!\cdots\!99}{18\!\cdots\!84}a^{8}-\frac{56\!\cdots\!71}{37\!\cdots\!68}a^{7}-\frac{63\!\cdots\!03}{37\!\cdots\!68}a^{6}+\frac{44\!\cdots\!77}{37\!\cdots\!68}a^{5}+\frac{71\!\cdots\!17}{18\!\cdots\!84}a^{4}-\frac{12\!\cdots\!63}{37\!\cdots\!68}a^{3}-\frac{11\!\cdots\!47}{94\!\cdots\!92}a^{2}+\frac{34\!\cdots\!49}{23\!\cdots\!48}a+\frac{16\!\cdots\!05}{47\!\cdots\!96}$, $\frac{48\!\cdots\!33}{37\!\cdots\!68}a^{17}-\frac{91\!\cdots\!93}{94\!\cdots\!92}a^{16}-\frac{19\!\cdots\!89}{18\!\cdots\!84}a^{15}+\frac{77\!\cdots\!37}{37\!\cdots\!68}a^{14}-\frac{27\!\cdots\!17}{18\!\cdots\!84}a^{13}-\frac{58\!\cdots\!59}{37\!\cdots\!68}a^{12}+\frac{15\!\cdots\!39}{94\!\cdots\!92}a^{11}+\frac{53\!\cdots\!47}{94\!\cdots\!92}a^{10}-\frac{55\!\cdots\!81}{94\!\cdots\!92}a^{9}-\frac{20\!\cdots\!43}{18\!\cdots\!84}a^{8}+\frac{34\!\cdots\!09}{37\!\cdots\!68}a^{7}+\frac{40\!\cdots\!61}{37\!\cdots\!68}a^{6}-\frac{22\!\cdots\!35}{37\!\cdots\!68}a^{5}-\frac{92\!\cdots\!59}{18\!\cdots\!84}a^{4}+\frac{54\!\cdots\!49}{37\!\cdots\!68}a^{3}+\frac{84\!\cdots\!09}{94\!\cdots\!92}a^{2}-\frac{13\!\cdots\!49}{23\!\cdots\!48}a-\frac{11\!\cdots\!91}{47\!\cdots\!96}$, $\frac{22\!\cdots\!39}{37\!\cdots\!68}a^{17}-\frac{53\!\cdots\!19}{94\!\cdots\!92}a^{16}+\frac{51\!\cdots\!33}{18\!\cdots\!84}a^{15}+\frac{44\!\cdots\!63}{37\!\cdots\!68}a^{14}-\frac{44\!\cdots\!03}{18\!\cdots\!84}a^{13}-\frac{32\!\cdots\!13}{37\!\cdots\!68}a^{12}+\frac{20\!\cdots\!21}{94\!\cdots\!92}a^{11}+\frac{30\!\cdots\!01}{94\!\cdots\!92}a^{10}-\frac{80\!\cdots\!43}{94\!\cdots\!92}a^{9}-\frac{14\!\cdots\!85}{18\!\cdots\!84}a^{8}+\frac{58\!\cdots\!51}{37\!\cdots\!68}a^{7}+\frac{43\!\cdots\!87}{37\!\cdots\!68}a^{6}-\frac{43\!\cdots\!93}{37\!\cdots\!68}a^{5}-\frac{14\!\cdots\!69}{18\!\cdots\!84}a^{4}+\frac{10\!\cdots\!79}{37\!\cdots\!68}a^{3}+\frac{16\!\cdots\!79}{94\!\cdots\!92}a^{2}-\frac{19\!\cdots\!43}{23\!\cdots\!48}a-\frac{25\!\cdots\!93}{47\!\cdots\!96}$, $\frac{76\!\cdots\!09}{18\!\cdots\!84}a^{17}-\frac{12\!\cdots\!97}{47\!\cdots\!96}a^{16}-\frac{57\!\cdots\!89}{94\!\cdots\!92}a^{15}+\frac{10\!\cdots\!85}{18\!\cdots\!84}a^{14}+\frac{16\!\cdots\!55}{94\!\cdots\!92}a^{13}-\frac{83\!\cdots\!75}{18\!\cdots\!84}a^{12}+\frac{47\!\cdots\!95}{47\!\cdots\!96}a^{11}+\frac{73\!\cdots\!99}{47\!\cdots\!96}a^{10}+\frac{52\!\cdots\!35}{47\!\cdots\!96}a^{9}-\frac{21\!\cdots\!15}{94\!\cdots\!92}a^{8}-\frac{12\!\cdots\!47}{18\!\cdots\!84}a^{7}+\frac{80\!\cdots\!33}{18\!\cdots\!84}a^{6}+\frac{51\!\cdots\!93}{18\!\cdots\!84}a^{5}+\frac{11\!\cdots\!69}{94\!\cdots\!92}a^{4}+\frac{73\!\cdots\!25}{18\!\cdots\!84}a^{3}-\frac{26\!\cdots\!95}{47\!\cdots\!96}a^{2}-\frac{10\!\cdots\!90}{58\!\cdots\!87}a+\frac{46\!\cdots\!49}{23\!\cdots\!48}$, $\frac{64\!\cdots\!87}{37\!\cdots\!68}a^{17}-\frac{10\!\cdots\!75}{94\!\cdots\!92}a^{16}-\frac{47\!\cdots\!79}{18\!\cdots\!84}a^{15}+\frac{94\!\cdots\!03}{37\!\cdots\!68}a^{14}+\frac{11\!\cdots\!37}{18\!\cdots\!84}a^{13}-\frac{78\!\cdots\!29}{37\!\cdots\!68}a^{12}+\frac{22\!\cdots\!73}{94\!\cdots\!92}a^{11}+\frac{79\!\cdots\!33}{94\!\cdots\!92}a^{10}-\frac{49\!\cdots\!55}{94\!\cdots\!92}a^{9}-\frac{32\!\cdots\!25}{18\!\cdots\!84}a^{8}-\frac{54\!\cdots\!09}{37\!\cdots\!68}a^{7}+\frac{64\!\cdots\!99}{37\!\cdots\!68}a^{6}+\frac{10\!\cdots\!59}{37\!\cdots\!68}a^{5}-\frac{14\!\cdots\!45}{18\!\cdots\!84}a^{4}-\frac{51\!\cdots\!37}{37\!\cdots\!68}a^{3}+\frac{11\!\cdots\!99}{94\!\cdots\!92}a^{2}+\frac{50\!\cdots\!19}{23\!\cdots\!48}a-\frac{12\!\cdots\!89}{47\!\cdots\!96}$, $\frac{58\!\cdots\!03}{37\!\cdots\!68}a^{17}-\frac{15\!\cdots\!83}{94\!\cdots\!92}a^{16}+\frac{30\!\cdots\!81}{18\!\cdots\!84}a^{15}+\frac{13\!\cdots\!67}{37\!\cdots\!68}a^{14}-\frac{16\!\cdots\!67}{18\!\cdots\!84}a^{13}-\frac{11\!\cdots\!85}{37\!\cdots\!68}a^{12}+\frac{76\!\cdots\!61}{94\!\cdots\!92}a^{11}+\frac{12\!\cdots\!01}{94\!\cdots\!92}a^{10}-\frac{30\!\cdots\!63}{94\!\cdots\!92}a^{9}-\frac{74\!\cdots\!29}{18\!\cdots\!84}a^{8}+\frac{22\!\cdots\!95}{37\!\cdots\!68}a^{7}+\frac{26\!\cdots\!31}{37\!\cdots\!68}a^{6}-\frac{14\!\cdots\!49}{37\!\cdots\!68}a^{5}-\frac{10\!\cdots\!37}{18\!\cdots\!84}a^{4}+\frac{20\!\cdots\!19}{37\!\cdots\!68}a^{3}+\frac{13\!\cdots\!39}{94\!\cdots\!92}a^{2}+\frac{27\!\cdots\!69}{23\!\cdots\!48}a-\frac{32\!\cdots\!97}{47\!\cdots\!96}$, $\frac{23\!\cdots\!01}{37\!\cdots\!68}a^{17}-\frac{32\!\cdots\!69}{94\!\cdots\!92}a^{16}-\frac{26\!\cdots\!33}{18\!\cdots\!84}a^{15}+\frac{32\!\cdots\!61}{37\!\cdots\!68}a^{14}+\frac{23\!\cdots\!39}{18\!\cdots\!84}a^{13}-\frac{29\!\cdots\!51}{37\!\cdots\!68}a^{12}-\frac{65\!\cdots\!05}{94\!\cdots\!92}a^{11}+\frac{32\!\cdots\!11}{94\!\cdots\!92}a^{10}+\frac{26\!\cdots\!59}{94\!\cdots\!92}a^{9}-\frac{14\!\cdots\!63}{18\!\cdots\!84}a^{8}-\frac{23\!\cdots\!35}{37\!\cdots\!68}a^{7}+\frac{28\!\cdots\!45}{37\!\cdots\!68}a^{6}+\frac{23\!\cdots\!21}{37\!\cdots\!68}a^{5}-\frac{63\!\cdots\!11}{18\!\cdots\!84}a^{4}-\frac{94\!\cdots\!75}{37\!\cdots\!68}a^{3}+\frac{64\!\cdots\!01}{94\!\cdots\!92}a^{2}+\frac{76\!\cdots\!41}{23\!\cdots\!48}a-\frac{26\!\cdots\!39}{47\!\cdots\!96}$, $\frac{14\!\cdots\!61}{18\!\cdots\!84}a^{17}-\frac{24\!\cdots\!25}{47\!\cdots\!96}a^{16}-\frac{97\!\cdots\!05}{94\!\cdots\!92}a^{15}+\frac{21\!\cdots\!57}{18\!\cdots\!84}a^{14}+\frac{47\!\cdots\!75}{94\!\cdots\!92}a^{13}-\frac{17\!\cdots\!87}{18\!\cdots\!84}a^{12}+\frac{13\!\cdots\!87}{47\!\cdots\!96}a^{11}+\frac{17\!\cdots\!31}{47\!\cdots\!96}a^{10}-\frac{43\!\cdots\!05}{47\!\cdots\!96}a^{9}-\frac{71\!\cdots\!55}{94\!\cdots\!92}a^{8}+\frac{13\!\cdots\!57}{18\!\cdots\!84}a^{7}+\frac{14\!\cdots\!69}{18\!\cdots\!84}a^{6}-\frac{43\!\cdots\!39}{18\!\cdots\!84}a^{5}-\frac{33\!\cdots\!55}{94\!\cdots\!92}a^{4}-\frac{38\!\cdots\!55}{18\!\cdots\!84}a^{3}+\frac{33\!\cdots\!77}{47\!\cdots\!96}a^{2}+\frac{85\!\cdots\!97}{11\!\cdots\!74}a-\frac{67\!\cdots\!11}{23\!\cdots\!48}$, $\frac{40\!\cdots\!27}{18\!\cdots\!84}a^{17}-\frac{12\!\cdots\!59}{47\!\cdots\!96}a^{16}-\frac{97\!\cdots\!51}{94\!\cdots\!92}a^{15}+\frac{25\!\cdots\!99}{18\!\cdots\!84}a^{14}+\frac{15\!\cdots\!61}{94\!\cdots\!92}a^{13}-\frac{35\!\cdots\!41}{18\!\cdots\!84}a^{12}-\frac{60\!\cdots\!87}{47\!\cdots\!96}a^{11}+\frac{41\!\cdots\!13}{47\!\cdots\!96}a^{10}+\frac{23\!\cdots\!57}{47\!\cdots\!96}a^{9}-\frac{10\!\cdots\!53}{94\!\cdots\!92}a^{8}-\frac{18\!\cdots\!61}{18\!\cdots\!84}a^{7}-\frac{33\!\cdots\!89}{18\!\cdots\!84}a^{6}+\frac{13\!\cdots\!99}{18\!\cdots\!84}a^{5}+\frac{28\!\cdots\!11}{94\!\cdots\!92}a^{4}-\frac{32\!\cdots\!73}{18\!\cdots\!84}a^{3}-\frac{53\!\cdots\!41}{47\!\cdots\!96}a^{2}-\frac{29\!\cdots\!65}{58\!\cdots\!87}a+\frac{11\!\cdots\!71}{23\!\cdots\!48}$, $\frac{16\!\cdots\!63}{37\!\cdots\!68}a^{17}-\frac{25\!\cdots\!07}{94\!\cdots\!92}a^{16}-\frac{12\!\cdots\!31}{18\!\cdots\!84}a^{15}+\frac{22\!\cdots\!91}{37\!\cdots\!68}a^{14}+\frac{53\!\cdots\!69}{18\!\cdots\!84}a^{13}-\frac{16\!\cdots\!13}{37\!\cdots\!68}a^{12}-\frac{77\!\cdots\!03}{94\!\cdots\!92}a^{11}+\frac{13\!\cdots\!21}{94\!\cdots\!92}a^{10}+\frac{50\!\cdots\!93}{94\!\cdots\!92}a^{9}-\frac{30\!\cdots\!57}{18\!\cdots\!84}a^{8}-\frac{54\!\cdots\!85}{37\!\cdots\!68}a^{7}-\frac{46\!\cdots\!41}{37\!\cdots\!68}a^{6}+\frac{19\!\cdots\!55}{37\!\cdots\!68}a^{5}+\frac{48\!\cdots\!39}{18\!\cdots\!84}a^{4}+\frac{22\!\cdots\!43}{37\!\cdots\!68}a^{3}-\frac{86\!\cdots\!53}{94\!\cdots\!92}a^{2}-\frac{56\!\cdots\!57}{23\!\cdots\!48}a+\frac{20\!\cdots\!63}{47\!\cdots\!96}$, $\frac{12\!\cdots\!07}{18\!\cdots\!84}a^{17}-\frac{17\!\cdots\!29}{47\!\cdots\!96}a^{16}-\frac{12\!\cdots\!87}{94\!\cdots\!92}a^{15}+\frac{16\!\cdots\!75}{18\!\cdots\!84}a^{14}+\frac{94\!\cdots\!57}{94\!\cdots\!92}a^{13}-\frac{14\!\cdots\!77}{18\!\cdots\!84}a^{12}-\frac{25\!\cdots\!57}{47\!\cdots\!96}a^{11}+\frac{14\!\cdots\!41}{47\!\cdots\!96}a^{10}+\frac{10\!\cdots\!01}{47\!\cdots\!96}a^{9}-\frac{53\!\cdots\!05}{94\!\cdots\!92}a^{8}-\frac{99\!\cdots\!85}{18\!\cdots\!84}a^{7}+\frac{69\!\cdots\!31}{18\!\cdots\!84}a^{6}+\frac{84\!\cdots\!91}{18\!\cdots\!84}a^{5}-\frac{29\!\cdots\!73}{94\!\cdots\!92}a^{4}-\frac{23\!\cdots\!09}{18\!\cdots\!84}a^{3}-\frac{10\!\cdots\!83}{47\!\cdots\!96}a^{2}+\frac{55\!\cdots\!83}{11\!\cdots\!74}a+\frac{99\!\cdots\!27}{23\!\cdots\!48}$, $\frac{39\!\cdots\!49}{18\!\cdots\!84}a^{17}-\frac{73\!\cdots\!23}{47\!\cdots\!96}a^{16}-\frac{15\!\cdots\!97}{94\!\cdots\!92}a^{15}+\frac{63\!\cdots\!37}{18\!\cdots\!84}a^{14}-\frac{21\!\cdots\!45}{94\!\cdots\!92}a^{13}-\frac{47\!\cdots\!71}{18\!\cdots\!84}a^{12}+\frac{12\!\cdots\!73}{47\!\cdots\!96}a^{11}+\frac{44\!\cdots\!39}{47\!\cdots\!96}a^{10}-\frac{45\!\cdots\!65}{47\!\cdots\!96}a^{9}-\frac{18\!\cdots\!87}{94\!\cdots\!92}a^{8}+\frac{28\!\cdots\!29}{18\!\cdots\!84}a^{7}+\frac{38\!\cdots\!65}{18\!\cdots\!84}a^{6}-\frac{19\!\cdots\!07}{18\!\cdots\!84}a^{5}-\frac{10\!\cdots\!03}{94\!\cdots\!92}a^{4}+\frac{47\!\cdots\!49}{18\!\cdots\!84}a^{3}+\frac{10\!\cdots\!23}{47\!\cdots\!96}a^{2}-\frac{78\!\cdots\!61}{11\!\cdots\!74}a-\frac{22\!\cdots\!59}{23\!\cdots\!48}$, $\frac{125920666707489}{37\!\cdots\!68}a^{17}+\frac{15\!\cdots\!79}{94\!\cdots\!92}a^{16}-\frac{21\!\cdots\!77}{18\!\cdots\!84}a^{15}-\frac{11\!\cdots\!43}{37\!\cdots\!68}a^{14}+\frac{49\!\cdots\!39}{18\!\cdots\!84}a^{13}+\frac{72\!\cdots\!77}{37\!\cdots\!68}a^{12}-\frac{22\!\cdots\!09}{94\!\cdots\!92}a^{11}-\frac{75\!\cdots\!81}{94\!\cdots\!92}a^{10}+\frac{10\!\cdots\!59}{94\!\cdots\!92}a^{9}+\frac{64\!\cdots\!45}{18\!\cdots\!84}a^{8}-\frac{91\!\cdots\!51}{37\!\cdots\!68}a^{7}-\frac{35\!\cdots\!11}{37\!\cdots\!68}a^{6}+\frac{97\!\cdots\!97}{37\!\cdots\!68}a^{5}+\frac{21\!\cdots\!53}{18\!\cdots\!84}a^{4}-\frac{42\!\cdots\!07}{37\!\cdots\!68}a^{3}-\frac{46\!\cdots\!11}{94\!\cdots\!92}a^{2}+\frac{38\!\cdots\!81}{23\!\cdots\!48}a+\frac{33\!\cdots\!57}{47\!\cdots\!96}$, $\frac{34\!\cdots\!39}{37\!\cdots\!68}a^{17}-\frac{40\!\cdots\!55}{94\!\cdots\!92}a^{16}-\frac{45\!\cdots\!71}{18\!\cdots\!84}a^{15}+\frac{40\!\cdots\!63}{37\!\cdots\!68}a^{14}+\frac{50\!\cdots\!53}{18\!\cdots\!84}a^{13}-\frac{36\!\cdots\!53}{37\!\cdots\!68}a^{12}-\frac{17\!\cdots\!35}{94\!\cdots\!92}a^{11}+\frac{37\!\cdots\!65}{94\!\cdots\!92}a^{10}+\frac{70\!\cdots\!13}{94\!\cdots\!92}a^{9}-\frac{12\!\cdots\!61}{18\!\cdots\!84}a^{8}-\frac{58\!\cdots\!65}{37\!\cdots\!68}a^{7}+\frac{84\!\cdots\!27}{37\!\cdots\!68}a^{6}+\frac{47\!\cdots\!27}{37\!\cdots\!68}a^{5}+\frac{40\!\cdots\!47}{18\!\cdots\!84}a^{4}-\frac{12\!\cdots\!65}{37\!\cdots\!68}a^{3}-\frac{92\!\cdots\!29}{94\!\cdots\!92}a^{2}+\frac{34\!\cdots\!17}{23\!\cdots\!48}a+\frac{21\!\cdots\!79}{47\!\cdots\!96}$, $\frac{31\!\cdots\!95}{37\!\cdots\!68}a^{17}-\frac{50\!\cdots\!51}{94\!\cdots\!92}a^{16}-\frac{22\!\cdots\!51}{18\!\cdots\!84}a^{15}+\frac{45\!\cdots\!07}{37\!\cdots\!68}a^{14}+\frac{57\!\cdots\!73}{18\!\cdots\!84}a^{13}-\frac{37\!\cdots\!53}{37\!\cdots\!68}a^{12}+\frac{85\!\cdots\!17}{94\!\cdots\!92}a^{11}+\frac{36\!\cdots\!93}{94\!\cdots\!92}a^{10}-\frac{51\!\cdots\!43}{94\!\cdots\!92}a^{9}-\frac{14\!\cdots\!17}{18\!\cdots\!84}a^{8}-\frac{52\!\cdots\!25}{37\!\cdots\!68}a^{7}+\frac{26\!\cdots\!43}{37\!\cdots\!68}a^{6}+\frac{91\!\cdots\!83}{37\!\cdots\!68}a^{5}-\frac{51\!\cdots\!81}{18\!\cdots\!84}a^{4}-\frac{55\!\cdots\!25}{37\!\cdots\!68}a^{3}+\frac{28\!\cdots\!31}{94\!\cdots\!92}a^{2}+\frac{74\!\cdots\!07}{23\!\cdots\!48}a+\frac{13\!\cdots\!51}{47\!\cdots\!96}$, $\frac{14\!\cdots\!29}{94\!\cdots\!92}a^{17}-\frac{27\!\cdots\!67}{23\!\cdots\!48}a^{16}-\frac{60\!\cdots\!85}{47\!\cdots\!96}a^{15}+\frac{23\!\cdots\!17}{94\!\cdots\!92}a^{14}-\frac{76\!\cdots\!13}{47\!\cdots\!96}a^{13}-\frac{17\!\cdots\!35}{94\!\cdots\!92}a^{12}+\frac{45\!\cdots\!57}{23\!\cdots\!48}a^{11}+\frac{16\!\cdots\!27}{23\!\cdots\!48}a^{10}-\frac{16\!\cdots\!45}{23\!\cdots\!48}a^{9}-\frac{67\!\cdots\!99}{47\!\cdots\!96}a^{8}+\frac{10\!\cdots\!01}{94\!\cdots\!92}a^{7}+\frac{14\!\cdots\!85}{94\!\cdots\!92}a^{6}-\frac{67\!\cdots\!07}{94\!\cdots\!92}a^{5}-\frac{37\!\cdots\!83}{47\!\cdots\!96}a^{4}+\frac{15\!\cdots\!65}{94\!\cdots\!92}a^{3}+\frac{40\!\cdots\!93}{23\!\cdots\!48}a^{2}-\frac{502934546769420}{58\!\cdots\!87}a-\frac{74\!\cdots\!37}{11\!\cdots\!74}$ 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:  \( 14711932756.8 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 14711932756.8 \cdot 1}{2\cdot\sqrt{16096057926792443193781494140625}}\cr\approx \mathstrut & 0.480639981584 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 18*x^16 + 141*x^15 + 108*x^14 - 1227*x^13 - 438*x^12 + 5220*x^11 + 1956*x^10 - 11198*x^9 - 5475*x^8 + 11283*x^7 + 6315*x^6 - 5184*x^5 - 3051*x^4 + 930*x^3 + 552*x^2 - 24*x - 16)
 
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 - 6*x^17 - 18*x^16 + 141*x^15 + 108*x^14 - 1227*x^13 - 438*x^12 + 5220*x^11 + 1956*x^10 - 11198*x^9 - 5475*x^8 + 11283*x^7 + 6315*x^6 - 5184*x^5 - 3051*x^4 + 930*x^3 + 552*x^2 - 24*x - 16, 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 - 6*x^17 - 18*x^16 + 141*x^15 + 108*x^14 - 1227*x^13 - 438*x^12 + 5220*x^11 + 1956*x^10 - 11198*x^9 - 5475*x^8 + 11283*x^7 + 6315*x^6 - 5184*x^5 - 3051*x^4 + 930*x^3 + 552*x^2 - 24*x - 16);
 
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 - 6*x^17 - 18*x^16 + 141*x^15 + 108*x^14 - 1227*x^13 - 438*x^12 + 5220*x^11 + 1956*x^10 - 11198*x^9 - 5475*x^8 + 11283*x^7 + 6315*x^6 - 5184*x^5 - 3051*x^4 + 930*x^3 + 552*x^2 - 24*x - 16);
 
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

$\He_3:C_4$ (as 18T49):

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 108
The 14 conjugacy class representatives for $\He_3:C_4$
Character table for $\He_3:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 6.6.55130625.1

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]
 

Sibling fields

Degree 18 sibling: data not computed
Degree 27 sibling: data not computed
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 ${\href{/padicField/2.4.0.1}{4} }^{3}{,}\,{\href{/padicField/2.2.0.1}{2} }^{3}$ R R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ R ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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 Deg $18$$9$$2$$26$
\(5\) Copy content Toggle raw display 5.6.3.1$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.9.1$x^{12} - 30 x^{8} + 225 x^{4} + 1125$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(11\) Copy content Toggle raw display 11.3.0.1$x^{3} + 2 x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.4.2$x^{6} - 110 x^{3} - 16819$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.9.6.1$x^{9} + 6 x^{7} + 60 x^{6} + 12 x^{5} + 42 x^{4} - 1465 x^{3} + 240 x^{2} - 1560 x + 8088$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$