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Properties

Label 18.18.5629084797...5024.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{33}\cdot 47^{11}$
Root discriminant $125.10$
Ramified primes $2, 3, 47$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3:S_4$ (as 18T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22115221, -36169521, -48626967, 78699796, 36211158, -58701558, -11912402, 21086784, 1648491, -4078547, -14037, 438816, -19822, -25806, 1986, 768, -75, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 75*x^16 + 768*x^15 + 1986*x^14 - 25806*x^13 - 19822*x^12 + 438816*x^11 - 14037*x^10 - 4078547*x^9 + 1648491*x^8 + 21086784*x^7 - 11912402*x^6 - 58701558*x^5 + 36211158*x^4 + 78699796*x^3 - 48626967*x^2 - 36169521*x + 22115221)
 
gp: K = bnfinit(x^18 - 9*x^17 - 75*x^16 + 768*x^15 + 1986*x^14 - 25806*x^13 - 19822*x^12 + 438816*x^11 - 14037*x^10 - 4078547*x^9 + 1648491*x^8 + 21086784*x^7 - 11912402*x^6 - 58701558*x^5 + 36211158*x^4 + 78699796*x^3 - 48626967*x^2 - 36169521*x + 22115221, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 75 x^{16} + 768 x^{15} + 1986 x^{14} - 25806 x^{13} - 19822 x^{12} + 438816 x^{11} - 14037 x^{10} - 4078547 x^{9} + 1648491 x^{8} + 21086784 x^{7} - 11912402 x^{6} - 58701558 x^{5} + 36211158 x^{4} + 78699796 x^{3} - 48626967 x^{2} - 36169521 x + 22115221 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(56290847976736302998762533947799425024=2^{12}\cdot 3^{33}\cdot 47^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.10$
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:  $2, 3, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{24} a^{15} - \frac{1}{8} a^{13} - \frac{1}{24} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{24} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{11}{24}$, $\frac{1}{528} a^{16} - \frac{1}{264} a^{15} + \frac{1}{88} a^{14} - \frac{5}{66} a^{13} + \frac{2}{33} a^{12} + \frac{9}{88} a^{11} - \frac{3}{44} a^{10} + \frac{7}{88} a^{9} - \frac{1}{176} a^{8} - \frac{1}{264} a^{7} - \frac{23}{132} a^{6} - \frac{21}{88} a^{5} + \frac{9}{22} a^{4} - \frac{1}{2} a^{3} - \frac{13}{88} a^{2} - \frac{103}{264} a - \frac{23}{528}$, $\frac{1}{385866985135724564594702719844661980195083242672} a^{17} + \frac{5053746928049927159285048438328518083680037}{35078816830520414963154792713151089108643931152} a^{16} - \frac{1104994227479431126088547260469529395324994963}{192933492567862282297351359922330990097541621336} a^{15} + \frac{2120279078774497359844612618109421429304900091}{96466746283931141148675679961165495048770810668} a^{14} - \frac{240091826106132086751078493419638570599028059}{48233373141965570574337839980582747524385405334} a^{13} + \frac{1629746200337936734043882055341131929499828859}{24116686570982785287168919990291373762192702667} a^{12} + \frac{3655261075862260034277951747022682616248899633}{32155582094643713716225226653721831682923603556} a^{11} - \frac{1899503961321618238498501730042904955883257711}{21437054729762475810816817769147887788615735704} a^{10} - \frac{2246475988958272793576858545002101214266186357}{128622328378574854864900906614887326731694414224} a^{9} - \frac{18079150615753473638984302756323086934338144159}{385866985135724564594702719844661980195083242672} a^{8} - \frac{11012389927983799524726626098029942764784129775}{96466746283931141148675679961165495048770810668} a^{7} - \frac{11611724078743287901290338743103994547911173551}{192933492567862282297351359922330990097541621336} a^{6} - \frac{4448933184721847360050339170190353645688703903}{10718527364881237905408408884573943894307867852} a^{5} - \frac{24207938870180774861796453308269530581297820239}{64311164189287427432450453307443663365847207112} a^{4} + \frac{23816842155250529971561650235755216795590269109}{64311164189287427432450453307443663365847207112} a^{3} + \frac{73875605523601309406323905112585921103757992093}{192933492567862282297351359922330990097541621336} a^{2} - \frac{170909676312356144175567300967958169220450929811}{385866985135724564594702719844661980195083242672} a + \frac{1692837201482998623660540627489216393419976415}{20308788691353924452352774728666420010267539088}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 15891448025500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_4$ (as 18T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3:S_4$
Character table for $C_3:S_4$

Intermediate fields

3.3.11421.1, 3.3.45684.2, 3.3.45684.1, 3.3.564.1, 6.6.44851536.1, 9.9.13443473060864064.2

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
$3$3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.12.22.67$x^{12} + 99 x^{11} + 117 x^{10} - 114 x^{9} - 81 x^{8} + 9 x^{7} - 15 x^{6} + 54 x^{5} - 108 x^{4} + 45 x^{3} - 81 x^{2} - 108 x - 72$$6$$2$$22$$D_6$$[5/2]_{2}^{2}$
47Data not computed