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Properties

Label 18.0.98827743405...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 5^{8}$
Root discriminant $21.53$
Ramified primes $2, 3, 5$
Class number $3$
Class group $[3]$
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 15, -3, -576, 2970, -8202, 15654, -22344, 24795, -22135, 16395, -10272, 5430, -2394, 870, -252, 57, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 57*x^16 - 252*x^15 + 870*x^14 - 2394*x^13 + 5430*x^12 - 10272*x^11 + 16395*x^10 - 22135*x^9 + 24795*x^8 - 22344*x^7 + 15654*x^6 - 8202*x^5 + 2970*x^4 - 576*x^3 - 3*x^2 + 15*x + 1)
 
gp: K = bnfinit(x^18 - 9*x^17 + 57*x^16 - 252*x^15 + 870*x^14 - 2394*x^13 + 5430*x^12 - 10272*x^11 + 16395*x^10 - 22135*x^9 + 24795*x^8 - 22344*x^7 + 15654*x^6 - 8202*x^5 + 2970*x^4 - 576*x^3 - 3*x^2 + 15*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 57 x^{16} - 252 x^{15} + 870 x^{14} - 2394 x^{13} + 5430 x^{12} - 10272 x^{11} + 16395 x^{10} - 22135 x^{9} + 24795 x^{8} - 22344 x^{7} + 15654 x^{6} - 8202 x^{5} + 2970 x^{4} - 576 x^{3} - 3 x^{2} + 15 x + 1 \)

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:  \(-988277434054315200000000=-\,2^{12}\cdot 3^{31}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.53$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{11} - \frac{1}{20} a^{8} + \frac{3}{20} a^{7} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{3}{20} a^{3} + \frac{1}{5} a^{2} - \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{11} - \frac{1}{20} a^{9} + \frac{1}{10} a^{8} + \frac{1}{20} a^{7} - \frac{1}{10} a^{6} - \frac{1}{4} a^{5} - \frac{1}{10} a^{4} - \frac{3}{20} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{40} a^{14} - \frac{1}{40} a^{13} - \frac{1}{8} a^{11} - \frac{1}{40} a^{10} - \frac{1}{20} a^{9} + \frac{3}{40} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{9}{20} a^{5} + \frac{19}{40} a^{4} + \frac{3}{40} a^{3} + \frac{1}{20} a^{2} + \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{40} a^{15} - \frac{1}{40} a^{13} - \frac{1}{40} a^{12} - \frac{3}{40} a^{10} + \frac{1}{40} a^{9} + \frac{1}{10} a^{8} - \frac{1}{5} a^{7} + \frac{1}{8} a^{6} - \frac{3}{40} a^{5} + \frac{3}{10} a^{4} - \frac{13}{40} a^{3} - \frac{7}{40} a^{2} + \frac{1}{20} a - \frac{7}{40}$, $\frac{1}{241120} a^{16} - \frac{1}{30140} a^{15} + \frac{513}{120560} a^{14} + \frac{2507}{120560} a^{13} - \frac{2209}{120560} a^{12} - \frac{7463}{120560} a^{11} + \frac{649}{10960} a^{10} - \frac{17}{1370} a^{9} - \frac{1979}{48224} a^{8} - \frac{1355}{24112} a^{7} - \frac{29797}{120560} a^{6} + \frac{199}{30140} a^{5} + \frac{15139}{60280} a^{4} + \frac{1839}{10960} a^{3} + \frac{5251}{15070} a^{2} + \frac{21557}{120560} a + \frac{13327}{241120}$, $\frac{1}{63414560} a^{17} + \frac{123}{63414560} a^{16} - \frac{10413}{2882480} a^{15} + \frac{12999}{3170728} a^{14} + \frac{167451}{7926820} a^{13} - \frac{205687}{15853640} a^{12} - \frac{274277}{15853640} a^{11} - \frac{226381}{2882480} a^{10} + \frac{5421201}{63414560} a^{9} - \frac{5613787}{63414560} a^{8} - \frac{741977}{15853640} a^{7} - \frac{6253531}{31707280} a^{6} - \frac{120969}{3170728} a^{5} - \frac{2830321}{6341456} a^{4} - \frac{8948061}{31707280} a^{3} - \frac{12728179}{31707280} a^{2} + \frac{23238909}{63414560} a - \frac{5129079}{12682912}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{42777}{792682} a^{17} + \frac{727209}{1585364} a^{16} - \frac{4628299}{1585364} a^{15} + \frac{40336125}{3170728} a^{14} - \frac{140023965}{3170728} a^{13} + \frac{193312951}{1585364} a^{12} - \frac{890749413}{3170728} a^{11} + \frac{155891805}{288248} a^{10} - \frac{703835155}{792682} a^{9} + \frac{3927198645}{3170728} a^{8} - \frac{4611131985}{3170728} a^{7} + \frac{4465233271}{3170728} a^{6} - \frac{434775168}{396341} a^{5} + \frac{2112340875}{3170728} a^{4} - \frac{945083815}{3170728} a^{3} + \frac{142155609}{1585364} a^{2} - \frac{33776715}{3170728} a - \frac{2702009}{3170728} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 88939.46667030624 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.573956280000.4

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

Sibling fields

Degree 9 siblings: data not computed
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 R R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$