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Properties

Label 18.0.52258772213...4107.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{39}\cdot 17^{12}\cdot 19^{12}$
Root discriminant $508.82$
Ramified primes $3, 17, 19$
Class number $177147$ (GRH)
Class group $[3, 3, 3, 3, 9, 9, 27]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17373979, 0, 0, 33589614, 0, 0, 224141118, 0, 0, 74482, 0, 0, 3849, 0, 0, -96, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 96*x^15 + 3849*x^12 + 74482*x^9 + 224141118*x^6 + 33589614*x^3 + 17373979)
 
gp: K = bnfinit(x^18 - 96*x^15 + 3849*x^12 + 74482*x^9 + 224141118*x^6 + 33589614*x^3 + 17373979, 1)
 

Normalized defining polynomial

\( x^{18} - 96 x^{15} + 3849 x^{12} + 74482 x^{9} + 224141118 x^{6} + 33589614 x^{3} + 17373979 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5225877221356918592091419443200727020071175574107=-\,3^{39}\cdot 17^{12}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $508.82$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 17, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{6} - \frac{2}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{18} a^{9} - \frac{1}{6} a^{6} + \frac{1}{3} a^{3} + \frac{5}{18}$, $\frac{1}{18} a^{10} - \frac{1}{18} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{7}{18} a - \frac{4}{9}$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{8} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{5}{18} a^{2} - \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{54} a^{12} - \frac{1}{54} a^{9} - \frac{1}{9} a^{6} - \frac{7}{54} a^{3} + \frac{11}{27}$, $\frac{1}{1998} a^{13} - \frac{20}{999} a^{10} - \frac{31}{666} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{175}{1998} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{215}{1998} a + \frac{4}{9}$, $\frac{1}{5994} a^{14} + \frac{1}{5994} a^{13} + \frac{1}{162} a^{12} + \frac{71}{5994} a^{11} + \frac{71}{5994} a^{10} - \frac{1}{162} a^{9} - \frac{34}{999} a^{8} - \frac{34}{999} a^{7} - \frac{4}{27} a^{6} - \frac{1285}{5994} a^{5} + \frac{713}{5994} a^{4} + \frac{29}{162} a^{3} + \frac{1280}{2997} a^{2} + \frac{281}{2997} a + \frac{29}{81}$, $\frac{1}{34565479885857990186} a^{15} - \frac{212471063605964537}{34565479885857990186} a^{12} + \frac{364326532387903013}{17282739942928995093} a^{9} + \frac{5706255703512743675}{34565479885857990186} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{5205150226739261101}{17282739942928995093} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{152626607025928843}{467101079538621489}$, $\frac{1}{241958359201005931302} a^{16} - \frac{39470663776845467}{241958359201005931302} a^{13} - \frac{3095681464194478387}{120979179600502965651} a^{10} + \frac{4979654024230443581}{241958359201005931302} a^{7} + \frac{1}{3} a^{5} + \frac{15031572937033224277}{120979179600502965651} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{12013599173670948967}{120979179600502965651} a - \frac{1}{3}$, $\frac{1}{62667215033060536207218} a^{17} + \frac{1857319006257519176}{31333607516530268103609} a^{14} - \frac{1675316520519712656041}{62667215033060536207218} a^{11} - \frac{61867700469741165067}{62667215033060536207218} a^{8} + \frac{1}{9} a^{6} + \frac{98969205126004574135}{62667215033060536207218} a^{5} + \frac{4}{9} a^{3} - \frac{12415539322711029092111}{31333607516530268103609} a^{2} - \frac{2}{9}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}\times C_{27}$, which has order $177147$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{8404129}{582705033561894} a^{15} + \frac{807537797}{582705033561894} a^{12} - \frac{32407177649}{582705033561894} a^{9} - \frac{328715016388}{291352516780947} a^{6} - \frac{941011222763530}{291352516780947} a^{3} + \frac{4062354848561}{15748784690862} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2807605647887.7075 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.25351947.3 x3, 6.0.14795494587.2, Deg 6, 6.0.2565108243.3, Deg 6

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{9}$ R R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$