Properties

Label 18.18.255...761.1
Degree $18$
Signature $[18, 0]$
Discriminant $2.560\times 10^{29}$
Root discriminant \(43.03\)
Ramified primes $19,31$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^2 : C_9$ (as 18T7)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 9*x^16 + 214*x^15 - 296*x^14 - 1586*x^13 + 3387*x^12 + 4963*x^11 - 12716*x^10 - 8725*x^9 + 21199*x^8 + 10625*x^7 - 15005*x^6 - 7039*x^5 + 2915*x^4 + 746*x^3 - 275*x^2 + 11*x + 1)
 
gp: K = bnfinit(y^18 - 8*y^17 - 9*y^16 + 214*y^15 - 296*y^14 - 1586*y^13 + 3387*y^12 + 4963*y^11 - 12716*y^10 - 8725*y^9 + 21199*y^8 + 10625*y^7 - 15005*y^6 - 7039*y^5 + 2915*y^4 + 746*y^3 - 275*y^2 + 11*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 8*x^17 - 9*x^16 + 214*x^15 - 296*x^14 - 1586*x^13 + 3387*x^12 + 4963*x^11 - 12716*x^10 - 8725*x^9 + 21199*x^8 + 10625*x^7 - 15005*x^6 - 7039*x^5 + 2915*x^4 + 746*x^3 - 275*x^2 + 11*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 8*x^17 - 9*x^16 + 214*x^15 - 296*x^14 - 1586*x^13 + 3387*x^12 + 4963*x^11 - 12716*x^10 - 8725*x^9 + 21199*x^8 + 10625*x^7 - 15005*x^6 - 7039*x^5 + 2915*x^4 + 746*x^3 - 275*x^2 + 11*x + 1)
 

\( x^{18} - 8 x^{17} - 9 x^{16} + 214 x^{15} - 296 x^{14} - 1586 x^{13} + 3387 x^{12} + 4963 x^{11} - 12716 x^{10} - 8725 x^{9} + 21199 x^{8} + 10625 x^{7} - 15005 x^{6} - 7039 x^{5} + \cdots + 1 \) 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:   \(255992816294107128730405733761\) \(\medspace = 19^{16}\cdot 31^{6}\) 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:  \(43.03\)
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:  $19^{8/9}31^{1/2}\approx 76.26946753496667$
Ramified primes:   \(19\), \(31\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{44\!\cdots\!09}a^{17}-\frac{72\!\cdots\!09}{44\!\cdots\!09}a^{16}+\frac{11\!\cdots\!96}{44\!\cdots\!09}a^{15}-\frac{39\!\cdots\!16}{44\!\cdots\!09}a^{14}+\frac{15\!\cdots\!38}{44\!\cdots\!09}a^{13}-\frac{18\!\cdots\!24}{44\!\cdots\!09}a^{12}-\frac{10\!\cdots\!92}{44\!\cdots\!09}a^{11}+\frac{49\!\cdots\!85}{44\!\cdots\!09}a^{10}+\frac{12\!\cdots\!22}{44\!\cdots\!09}a^{9}-\frac{17\!\cdots\!41}{44\!\cdots\!09}a^{8}-\frac{15\!\cdots\!44}{44\!\cdots\!09}a^{7}+\frac{13\!\cdots\!42}{44\!\cdots\!09}a^{6}+\frac{52\!\cdots\!06}{44\!\cdots\!09}a^{5}-\frac{45\!\cdots\!15}{44\!\cdots\!09}a^{4}-\frac{18\!\cdots\!15}{44\!\cdots\!09}a^{3}+\frac{76\!\cdots\!44}{44\!\cdots\!09}a^{2}-\frac{49\!\cdots\!16}{44\!\cdots\!09}a+\frac{18\!\cdots\!81}{44\!\cdots\!09}$ 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 $1$ (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:   $\frac{32\!\cdots\!15}{44\!\cdots\!09}a^{17}-\frac{25\!\cdots\!35}{44\!\cdots\!09}a^{16}-\frac{35\!\cdots\!83}{44\!\cdots\!09}a^{15}+\frac{69\!\cdots\!96}{44\!\cdots\!09}a^{14}-\frac{81\!\cdots\!08}{44\!\cdots\!09}a^{13}-\frac{53\!\cdots\!01}{44\!\cdots\!09}a^{12}+\frac{98\!\cdots\!23}{44\!\cdots\!09}a^{11}+\frac{18\!\cdots\!30}{44\!\cdots\!09}a^{10}-\frac{36\!\cdots\!43}{44\!\cdots\!09}a^{9}-\frac{37\!\cdots\!62}{44\!\cdots\!09}a^{8}+\frac{59\!\cdots\!88}{44\!\cdots\!09}a^{7}+\frac{49\!\cdots\!78}{44\!\cdots\!09}a^{6}-\frac{35\!\cdots\!04}{44\!\cdots\!09}a^{5}-\frac{32\!\cdots\!99}{44\!\cdots\!09}a^{4}+\frac{65\!\cdots\!50}{44\!\cdots\!09}a^{3}+\frac{29\!\cdots\!55}{44\!\cdots\!09}a^{2}-\frac{64\!\cdots\!98}{44\!\cdots\!09}a-\frac{34\!\cdots\!98}{44\!\cdots\!09}$, $\frac{41\!\cdots\!36}{44\!\cdots\!09}a^{17}-\frac{32\!\cdots\!67}{44\!\cdots\!09}a^{16}-\frac{42\!\cdots\!72}{44\!\cdots\!09}a^{15}+\frac{87\!\cdots\!86}{44\!\cdots\!09}a^{14}-\frac{10\!\cdots\!21}{44\!\cdots\!09}a^{13}-\frac{67\!\cdots\!98}{44\!\cdots\!09}a^{12}+\frac{12\!\cdots\!85}{44\!\cdots\!09}a^{11}+\frac{22\!\cdots\!64}{44\!\cdots\!09}a^{10}-\frac{47\!\cdots\!50}{44\!\cdots\!09}a^{9}-\frac{43\!\cdots\!61}{44\!\cdots\!09}a^{8}+\frac{76\!\cdots\!16}{44\!\cdots\!09}a^{7}+\frac{57\!\cdots\!85}{44\!\cdots\!09}a^{6}-\frac{46\!\cdots\!53}{44\!\cdots\!09}a^{5}-\frac{37\!\cdots\!20}{44\!\cdots\!09}a^{4}+\frac{23\!\cdots\!08}{44\!\cdots\!09}a^{3}+\frac{31\!\cdots\!15}{44\!\cdots\!09}a^{2}-\frac{24\!\cdots\!64}{44\!\cdots\!09}a-\frac{68\!\cdots\!79}{44\!\cdots\!09}$, $\frac{59\!\cdots\!45}{44\!\cdots\!09}a^{17}-\frac{46\!\cdots\!10}{44\!\cdots\!09}a^{16}-\frac{62\!\cdots\!40}{44\!\cdots\!09}a^{15}+\frac{12\!\cdots\!17}{44\!\cdots\!09}a^{14}-\frac{15\!\cdots\!53}{44\!\cdots\!09}a^{13}-\frac{96\!\cdots\!49}{44\!\cdots\!09}a^{12}+\frac{18\!\cdots\!26}{44\!\cdots\!09}a^{11}+\frac{32\!\cdots\!05}{44\!\cdots\!09}a^{10}-\frac{68\!\cdots\!31}{44\!\cdots\!09}a^{9}-\frac{64\!\cdots\!18}{44\!\cdots\!09}a^{8}+\frac{11\!\cdots\!88}{44\!\cdots\!09}a^{7}+\frac{84\!\cdots\!25}{44\!\cdots\!09}a^{6}-\frac{68\!\cdots\!01}{44\!\cdots\!09}a^{5}-\frac{55\!\cdots\!89}{44\!\cdots\!09}a^{4}+\frac{43\!\cdots\!55}{44\!\cdots\!09}a^{3}+\frac{50\!\cdots\!08}{44\!\cdots\!09}a^{2}-\frac{44\!\cdots\!72}{44\!\cdots\!09}a-\frac{21\!\cdots\!45}{44\!\cdots\!09}$, $\frac{19\!\cdots\!13}{44\!\cdots\!09}a^{17}-\frac{13\!\cdots\!79}{44\!\cdots\!09}a^{16}-\frac{30\!\cdots\!05}{44\!\cdots\!09}a^{15}+\frac{39\!\cdots\!68}{44\!\cdots\!09}a^{14}-\frac{22\!\cdots\!85}{44\!\cdots\!09}a^{13}-\frac{36\!\cdots\!86}{44\!\cdots\!09}a^{12}+\frac{42\!\cdots\!32}{44\!\cdots\!09}a^{11}+\frac{15\!\cdots\!71}{44\!\cdots\!09}a^{10}-\frac{18\!\cdots\!28}{44\!\cdots\!09}a^{9}-\frac{36\!\cdots\!35}{44\!\cdots\!09}a^{8}+\frac{33\!\cdots\!26}{44\!\cdots\!09}a^{7}+\frac{49\!\cdots\!69}{44\!\cdots\!09}a^{6}-\frac{20\!\cdots\!20}{44\!\cdots\!09}a^{5}-\frac{30\!\cdots\!14}{44\!\cdots\!09}a^{4}-\frac{86\!\cdots\!53}{44\!\cdots\!09}a^{3}+\frac{26\!\cdots\!29}{44\!\cdots\!09}a^{2}-\frac{79\!\cdots\!21}{44\!\cdots\!09}a-\frac{12\!\cdots\!47}{44\!\cdots\!09}$, $\frac{10\!\cdots\!96}{44\!\cdots\!09}a^{17}-\frac{71\!\cdots\!55}{44\!\cdots\!09}a^{16}-\frac{16\!\cdots\!78}{44\!\cdots\!09}a^{15}+\frac{20\!\cdots\!54}{44\!\cdots\!09}a^{14}-\frac{10\!\cdots\!83}{44\!\cdots\!09}a^{13}-\frac{19\!\cdots\!94}{44\!\cdots\!09}a^{12}+\frac{21\!\cdots\!78}{44\!\cdots\!09}a^{11}+\frac{83\!\cdots\!49}{44\!\cdots\!09}a^{10}-\frac{97\!\cdots\!59}{44\!\cdots\!09}a^{9}-\frac{19\!\cdots\!79}{44\!\cdots\!09}a^{8}+\frac{17\!\cdots\!95}{44\!\cdots\!09}a^{7}+\frac{26\!\cdots\!54}{44\!\cdots\!09}a^{6}-\frac{10\!\cdots\!13}{44\!\cdots\!09}a^{5}-\frac{16\!\cdots\!02}{44\!\cdots\!09}a^{4}-\frac{76\!\cdots\!02}{44\!\cdots\!09}a^{3}+\frac{13\!\cdots\!70}{44\!\cdots\!09}a^{2}-\frac{51\!\cdots\!66}{44\!\cdots\!09}a-\frac{54\!\cdots\!89}{44\!\cdots\!09}$, $\frac{89\!\cdots\!17}{44\!\cdots\!09}a^{17}-\frac{64\!\cdots\!24}{44\!\cdots\!09}a^{16}-\frac{13\!\cdots\!27}{44\!\cdots\!09}a^{15}+\frac{18\!\cdots\!14}{44\!\cdots\!09}a^{14}-\frac{11\!\cdots\!02}{44\!\cdots\!09}a^{13}-\frac{16\!\cdots\!92}{44\!\cdots\!09}a^{12}+\frac{20\!\cdots\!54}{44\!\cdots\!09}a^{11}+\frac{70\!\cdots\!22}{44\!\cdots\!09}a^{10}-\frac{91\!\cdots\!69}{44\!\cdots\!09}a^{9}-\frac{16\!\cdots\!56}{44\!\cdots\!09}a^{8}+\frac{16\!\cdots\!31}{44\!\cdots\!09}a^{7}+\frac{22\!\cdots\!15}{44\!\cdots\!09}a^{6}-\frac{10\!\cdots\!07}{44\!\cdots\!09}a^{5}-\frac{13\!\cdots\!12}{44\!\cdots\!09}a^{4}-\frac{10\!\cdots\!51}{44\!\cdots\!09}a^{3}+\frac{12\!\cdots\!59}{44\!\cdots\!09}a^{2}-\frac{28\!\cdots\!55}{44\!\cdots\!09}a-\frac{16\!\cdots\!76}{44\!\cdots\!09}$, $\frac{41\!\cdots\!36}{44\!\cdots\!09}a^{17}-\frac{32\!\cdots\!67}{44\!\cdots\!09}a^{16}-\frac{42\!\cdots\!72}{44\!\cdots\!09}a^{15}+\frac{87\!\cdots\!86}{44\!\cdots\!09}a^{14}-\frac{10\!\cdots\!21}{44\!\cdots\!09}a^{13}-\frac{67\!\cdots\!98}{44\!\cdots\!09}a^{12}+\frac{12\!\cdots\!85}{44\!\cdots\!09}a^{11}+\frac{22\!\cdots\!64}{44\!\cdots\!09}a^{10}-\frac{47\!\cdots\!50}{44\!\cdots\!09}a^{9}-\frac{43\!\cdots\!61}{44\!\cdots\!09}a^{8}+\frac{76\!\cdots\!16}{44\!\cdots\!09}a^{7}+\frac{57\!\cdots\!85}{44\!\cdots\!09}a^{6}-\frac{46\!\cdots\!53}{44\!\cdots\!09}a^{5}-\frac{37\!\cdots\!20}{44\!\cdots\!09}a^{4}+\frac{23\!\cdots\!08}{44\!\cdots\!09}a^{3}+\frac{31\!\cdots\!15}{44\!\cdots\!09}a^{2}-\frac{24\!\cdots\!64}{44\!\cdots\!09}a-\frac{11\!\cdots\!88}{44\!\cdots\!09}$, $\frac{89\!\cdots\!17}{44\!\cdots\!09}a^{17}-\frac{64\!\cdots\!24}{44\!\cdots\!09}a^{16}-\frac{13\!\cdots\!27}{44\!\cdots\!09}a^{15}+\frac{18\!\cdots\!14}{44\!\cdots\!09}a^{14}-\frac{11\!\cdots\!02}{44\!\cdots\!09}a^{13}-\frac{16\!\cdots\!92}{44\!\cdots\!09}a^{12}+\frac{20\!\cdots\!54}{44\!\cdots\!09}a^{11}+\frac{70\!\cdots\!22}{44\!\cdots\!09}a^{10}-\frac{91\!\cdots\!69}{44\!\cdots\!09}a^{9}-\frac{16\!\cdots\!56}{44\!\cdots\!09}a^{8}+\frac{16\!\cdots\!31}{44\!\cdots\!09}a^{7}+\frac{22\!\cdots\!15}{44\!\cdots\!09}a^{6}-\frac{10\!\cdots\!07}{44\!\cdots\!09}a^{5}-\frac{13\!\cdots\!12}{44\!\cdots\!09}a^{4}-\frac{10\!\cdots\!51}{44\!\cdots\!09}a^{3}+\frac{12\!\cdots\!59}{44\!\cdots\!09}a^{2}-\frac{28\!\cdots\!55}{44\!\cdots\!09}a-\frac{11\!\cdots\!67}{44\!\cdots\!09}$, $\frac{89\!\cdots\!17}{44\!\cdots\!09}a^{17}-\frac{64\!\cdots\!24}{44\!\cdots\!09}a^{16}-\frac{13\!\cdots\!27}{44\!\cdots\!09}a^{15}+\frac{18\!\cdots\!14}{44\!\cdots\!09}a^{14}-\frac{11\!\cdots\!02}{44\!\cdots\!09}a^{13}-\frac{16\!\cdots\!92}{44\!\cdots\!09}a^{12}+\frac{20\!\cdots\!54}{44\!\cdots\!09}a^{11}+\frac{70\!\cdots\!22}{44\!\cdots\!09}a^{10}-\frac{91\!\cdots\!69}{44\!\cdots\!09}a^{9}-\frac{16\!\cdots\!56}{44\!\cdots\!09}a^{8}+\frac{16\!\cdots\!31}{44\!\cdots\!09}a^{7}+\frac{22\!\cdots\!15}{44\!\cdots\!09}a^{6}-\frac{10\!\cdots\!07}{44\!\cdots\!09}a^{5}-\frac{13\!\cdots\!12}{44\!\cdots\!09}a^{4}-\frac{10\!\cdots\!51}{44\!\cdots\!09}a^{3}+\frac{12\!\cdots\!59}{44\!\cdots\!09}a^{2}-\frac{32\!\cdots\!64}{44\!\cdots\!09}a-\frac{11\!\cdots\!67}{44\!\cdots\!09}$, $\frac{36\!\cdots\!80}{44\!\cdots\!09}a^{17}-\frac{29\!\cdots\!68}{44\!\cdots\!09}a^{16}-\frac{34\!\cdots\!75}{44\!\cdots\!09}a^{15}+\frac{77\!\cdots\!96}{44\!\cdots\!09}a^{14}-\frac{10\!\cdots\!42}{44\!\cdots\!09}a^{13}-\frac{57\!\cdots\!27}{44\!\cdots\!09}a^{12}+\frac{11\!\cdots\!18}{44\!\cdots\!09}a^{11}+\frac{18\!\cdots\!68}{44\!\cdots\!09}a^{10}-\frac{42\!\cdots\!73}{44\!\cdots\!09}a^{9}-\frac{34\!\cdots\!82}{44\!\cdots\!09}a^{8}+\frac{67\!\cdots\!39}{44\!\cdots\!09}a^{7}+\frac{44\!\cdots\!86}{44\!\cdots\!09}a^{6}-\frac{40\!\cdots\!68}{44\!\cdots\!09}a^{5}-\frac{29\!\cdots\!12}{44\!\cdots\!09}a^{4}+\frac{21\!\cdots\!19}{44\!\cdots\!09}a^{3}+\frac{24\!\cdots\!14}{44\!\cdots\!09}a^{2}-\frac{20\!\cdots\!84}{44\!\cdots\!09}a-\frac{84\!\cdots\!14}{44\!\cdots\!09}$, $\frac{24\!\cdots\!97}{44\!\cdots\!09}a^{17}-\frac{21\!\cdots\!83}{44\!\cdots\!09}a^{16}-\frac{45\!\cdots\!93}{44\!\cdots\!09}a^{15}+\frac{50\!\cdots\!66}{44\!\cdots\!09}a^{14}-\frac{11\!\cdots\!13}{44\!\cdots\!09}a^{13}-\frac{26\!\cdots\!32}{44\!\cdots\!09}a^{12}+\frac{96\!\cdots\!64}{44\!\cdots\!09}a^{11}+\frac{25\!\cdots\!67}{44\!\cdots\!09}a^{10}-\frac{27\!\cdots\!46}{44\!\cdots\!09}a^{9}+\frac{57\!\cdots\!32}{44\!\cdots\!09}a^{8}+\frac{28\!\cdots\!69}{44\!\cdots\!09}a^{7}-\frac{68\!\cdots\!73}{44\!\cdots\!09}a^{6}-\frac{37\!\cdots\!20}{44\!\cdots\!09}a^{5}+\frac{98\!\cdots\!53}{44\!\cdots\!09}a^{4}-\frac{93\!\cdots\!20}{44\!\cdots\!09}a^{3}-\frac{25\!\cdots\!63}{44\!\cdots\!09}a^{2}+\frac{85\!\cdots\!86}{44\!\cdots\!09}a+\frac{30\!\cdots\!11}{44\!\cdots\!09}$, $\frac{27\!\cdots\!52}{44\!\cdots\!09}a^{17}-\frac{20\!\cdots\!16}{44\!\cdots\!09}a^{16}-\frac{34\!\cdots\!99}{44\!\cdots\!09}a^{15}+\frac{57\!\cdots\!29}{44\!\cdots\!09}a^{14}-\frac{52\!\cdots\!83}{44\!\cdots\!09}a^{13}-\frac{47\!\cdots\!25}{44\!\cdots\!09}a^{12}+\frac{72\!\cdots\!16}{44\!\cdots\!09}a^{11}+\frac{17\!\cdots\!55}{44\!\cdots\!09}a^{10}-\frac{29\!\cdots\!58}{44\!\cdots\!09}a^{9}-\frac{39\!\cdots\!37}{44\!\cdots\!09}a^{8}+\frac{49\!\cdots\!20}{44\!\cdots\!09}a^{7}+\frac{52\!\cdots\!25}{44\!\cdots\!09}a^{6}-\frac{30\!\cdots\!55}{44\!\cdots\!09}a^{5}-\frac{33\!\cdots\!20}{44\!\cdots\!09}a^{4}+\frac{38\!\cdots\!63}{44\!\cdots\!09}a^{3}+\frac{30\!\cdots\!42}{44\!\cdots\!09}a^{2}-\frac{18\!\cdots\!16}{44\!\cdots\!09}a-\frac{15\!\cdots\!45}{44\!\cdots\!09}$, $a$, $\frac{34\!\cdots\!71}{44\!\cdots\!09}a^{17}-\frac{27\!\cdots\!58}{44\!\cdots\!09}a^{16}-\frac{32\!\cdots\!14}{44\!\cdots\!09}a^{15}+\frac{73\!\cdots\!75}{44\!\cdots\!09}a^{14}-\frac{97\!\cdots\!96}{44\!\cdots\!09}a^{13}-\frac{56\!\cdots\!77}{44\!\cdots\!09}a^{12}+\frac{11\!\cdots\!59}{44\!\cdots\!09}a^{11}+\frac{18\!\cdots\!99}{44\!\cdots\!09}a^{10}-\frac{45\!\cdots\!93}{44\!\cdots\!09}a^{9}-\frac{32\!\cdots\!08}{44\!\cdots\!09}a^{8}+\frac{77\!\cdots\!41}{44\!\cdots\!09}a^{7}+\frac{38\!\cdots\!17}{44\!\cdots\!09}a^{6}-\frac{58\!\cdots\!51}{44\!\cdots\!09}a^{5}-\frac{24\!\cdots\!88}{44\!\cdots\!09}a^{4}+\frac{13\!\cdots\!16}{44\!\cdots\!09}a^{3}+\frac{28\!\cdots\!75}{44\!\cdots\!09}a^{2}-\frac{12\!\cdots\!20}{44\!\cdots\!09}a+\frac{54\!\cdots\!99}{44\!\cdots\!09}$, $\frac{22\!\cdots\!64}{44\!\cdots\!09}a^{17}-\frac{18\!\cdots\!58}{44\!\cdots\!09}a^{16}-\frac{83\!\cdots\!50}{44\!\cdots\!09}a^{15}+\frac{47\!\cdots\!61}{44\!\cdots\!09}a^{14}-\frac{92\!\cdots\!19}{44\!\cdots\!09}a^{13}-\frac{28\!\cdots\!84}{44\!\cdots\!09}a^{12}+\frac{87\!\cdots\!20}{44\!\cdots\!09}a^{11}+\frac{50\!\cdots\!20}{44\!\cdots\!09}a^{10}-\frac{28\!\cdots\!76}{44\!\cdots\!09}a^{9}-\frac{17\!\cdots\!52}{44\!\cdots\!09}a^{8}+\frac{38\!\cdots\!38}{44\!\cdots\!09}a^{7}+\frac{12\!\cdots\!90}{44\!\cdots\!09}a^{6}-\frac{21\!\cdots\!28}{44\!\cdots\!09}a^{5}-\frac{35\!\cdots\!94}{44\!\cdots\!09}a^{4}+\frac{17\!\cdots\!19}{44\!\cdots\!09}a^{3}+\frac{19\!\cdots\!03}{44\!\cdots\!09}a^{2}-\frac{29\!\cdots\!25}{44\!\cdots\!09}a-\frac{11\!\cdots\!76}{44\!\cdots\!09}$, $\frac{91\!\cdots\!82}{44\!\cdots\!09}a^{17}-\frac{71\!\cdots\!21}{44\!\cdots\!09}a^{16}-\frac{96\!\cdots\!68}{44\!\cdots\!09}a^{15}+\frac{19\!\cdots\!52}{44\!\cdots\!09}a^{14}-\frac{23\!\cdots\!55}{44\!\cdots\!09}a^{13}-\frac{14\!\cdots\!10}{44\!\cdots\!09}a^{12}+\frac{27\!\cdots\!66}{44\!\cdots\!09}a^{11}+\frac{50\!\cdots\!49}{44\!\cdots\!09}a^{10}-\frac{10\!\cdots\!67}{44\!\cdots\!09}a^{9}-\frac{99\!\cdots\!86}{44\!\cdots\!09}a^{8}+\frac{17\!\cdots\!59}{44\!\cdots\!09}a^{7}+\frac{13\!\cdots\!05}{44\!\cdots\!09}a^{6}-\frac{10\!\cdots\!17}{44\!\cdots\!09}a^{5}-\frac{84\!\cdots\!21}{44\!\cdots\!09}a^{4}+\frac{75\!\cdots\!87}{44\!\cdots\!09}a^{3}+\frac{73\!\cdots\!77}{44\!\cdots\!09}a^{2}-\frac{81\!\cdots\!66}{44\!\cdots\!09}a-\frac{18\!\cdots\!13}{44\!\cdots\!09}$, $\frac{30\!\cdots\!13}{44\!\cdots\!09}a^{17}-\frac{31\!\cdots\!23}{44\!\cdots\!09}a^{16}+\frac{36\!\cdots\!57}{44\!\cdots\!09}a^{15}+\frac{66\!\cdots\!21}{44\!\cdots\!09}a^{14}-\frac{24\!\cdots\!14}{44\!\cdots\!09}a^{13}-\frac{15\!\cdots\!62}{44\!\cdots\!09}a^{12}+\frac{19\!\cdots\!11}{44\!\cdots\!09}a^{11}-\frac{15\!\cdots\!72}{44\!\cdots\!09}a^{10}-\frac{51\!\cdots\!87}{44\!\cdots\!09}a^{9}+\frac{71\!\cdots\!35}{44\!\cdots\!09}a^{8}+\frac{56\!\cdots\!24}{44\!\cdots\!09}a^{7}-\frac{10\!\cdots\!12}{44\!\cdots\!09}a^{6}-\frac{27\!\cdots\!88}{44\!\cdots\!09}a^{5}+\frac{54\!\cdots\!30}{44\!\cdots\!09}a^{4}+\frac{71\!\cdots\!29}{44\!\cdots\!09}a^{3}-\frac{60\!\cdots\!00}{44\!\cdots\!09}a^{2}+\frac{32\!\cdots\!53}{44\!\cdots\!09}a+\frac{21\!\cdots\!86}{44\!\cdots\!09}$ 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:  \( 618241654.691 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 618241654.691 \cdot 1}{2\cdot\sqrt{255992816294107128730405733761}}\cr\approx \mathstrut & 0.160160088432 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 9*x^16 + 214*x^15 - 296*x^14 - 1586*x^13 + 3387*x^12 + 4963*x^11 - 12716*x^10 - 8725*x^9 + 21199*x^8 + 10625*x^7 - 15005*x^6 - 7039*x^5 + 2915*x^4 + 746*x^3 - 275*x^2 + 11*x + 1)
 
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 - 8*x^17 - 9*x^16 + 214*x^15 - 296*x^14 - 1586*x^13 + 3387*x^12 + 4963*x^11 - 12716*x^10 - 8725*x^9 + 21199*x^8 + 10625*x^7 - 15005*x^6 - 7039*x^5 + 2915*x^4 + 746*x^3 - 275*x^2 + 11*x + 1, 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 - 8*x^17 - 9*x^16 + 214*x^15 - 296*x^14 - 1586*x^13 + 3387*x^12 + 4963*x^11 - 12716*x^10 - 8725*x^9 + 21199*x^8 + 10625*x^7 - 15005*x^6 - 7039*x^5 + 2915*x^4 + 746*x^3 - 275*x^2 + 11*x + 1);
 
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 - 8*x^17 - 9*x^16 + 214*x^15 - 296*x^14 - 1586*x^13 + 3387*x^12 + 4963*x^11 - 12716*x^10 - 8725*x^9 + 21199*x^8 + 10625*x^7 - 15005*x^6 - 7039*x^5 + 2915*x^4 + 746*x^3 - 275*x^2 + 11*x + 1);
 
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

$C_2^2:C_9$ (as 18T7):

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 36
The 12 conjugacy class representatives for $C_2^2 : C_9$
Character table for $C_2^2 : C_9$

Intermediate fields

3.3.361.1, 6.6.125238481.1, \(\Q(\zeta_{19})^+\)

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

Galois closure: 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 ${\href{/padicField/2.9.0.1}{9} }^{2}$ ${\href{/padicField/3.9.0.1}{9} }^{2}$ ${\href{/padicField/5.9.0.1}{9} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.9.0.1}{9} }^{2}$ R ${\href{/padicField/23.9.0.1}{9} }^{2}$ ${\href{/padicField/29.9.0.1}{9} }^{2}$ R ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ ${\href{/padicField/47.9.0.1}{9} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }^{2}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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 19.9.8.8$x^{9} + 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} + 19$$9$$1$$8$$C_9$$[\ ]_{9}$
\(31\) Copy content Toggle raw display 31.6.3.2$x^{6} + 95 x^{4} + 56 x^{3} + 2884 x^{2} - 5152 x + 28684$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.0.1$x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
31.6.3.1$x^{6} + 961 x^{2} - 834148$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$