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Properties

Label 18.0.31607550078...2927.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 47^{9}$
Root discriminant $29.66$
Ramified primes $3, 47$
Class number $15$
Class group $[15]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20331, 7938, 12069, 5976, 13032, 3024, 287, 900, 4833, 664, -435, 66, 459, -96, -48, 20, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 6*x^16 + 20*x^15 - 48*x^14 - 96*x^13 + 459*x^12 + 66*x^11 - 435*x^10 + 664*x^9 + 4833*x^8 + 900*x^7 + 287*x^6 + 3024*x^5 + 13032*x^4 + 5976*x^3 + 12069*x^2 + 7938*x + 20331)
 
gp: K = bnfinit(x^18 - 6*x^17 + 6*x^16 + 20*x^15 - 48*x^14 - 96*x^13 + 459*x^12 + 66*x^11 - 435*x^10 + 664*x^9 + 4833*x^8 + 900*x^7 + 287*x^6 + 3024*x^5 + 13032*x^4 + 5976*x^3 + 12069*x^2 + 7938*x + 20331, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 6 x^{16} + 20 x^{15} - 48 x^{14} - 96 x^{13} + 459 x^{12} + 66 x^{11} - 435 x^{10} + 664 x^{9} + 4833 x^{8} + 900 x^{7} + 287 x^{6} + 3024 x^{5} + 13032 x^{4} + 5976 x^{3} + 12069 x^{2} + 7938 x + 20331 \)

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:  \(-316075500780176721688542927=-\,3^{24}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.66$
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:  $3, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{11} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{108} a^{12} - \frac{1}{18} a^{11} - \frac{1}{12} a^{10} - \frac{2}{27} a^{9} + \frac{1}{36} a^{8} + \frac{1}{18} a^{7} + \frac{17}{108} a^{6} + \frac{1}{3} a^{5} + \frac{17}{36} a^{4} - \frac{23}{54} a^{3} - \frac{1}{4} a^{2} - \frac{1}{6} a - \frac{1}{12}$, $\frac{1}{108} a^{13} - \frac{1}{12} a^{11} - \frac{2}{27} a^{10} - \frac{1}{12} a^{9} + \frac{1}{18} a^{8} - \frac{1}{108} a^{7} - \frac{2}{9} a^{6} - \frac{13}{36} a^{5} - \frac{5}{54} a^{4} - \frac{5}{36} a^{3} - \frac{1}{2} a^{2} - \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{108} a^{14} - \frac{2}{27} a^{11} + \frac{1}{18} a^{9} + \frac{2}{27} a^{8} - \frac{1}{18} a^{7} + \frac{2}{9} a^{6} + \frac{2}{27} a^{5} + \frac{4}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{1296} a^{15} - \frac{1}{1296} a^{14} + \frac{1}{1296} a^{13} - \frac{1}{648} a^{12} - \frac{55}{1296} a^{11} + \frac{53}{648} a^{10} + \frac{35}{1296} a^{9} - \frac{13}{648} a^{8} + \frac{101}{1296} a^{7} - \frac{35}{162} a^{6} + \frac{55}{1296} a^{5} + \frac{2}{81} a^{4} + \frac{149}{432} a^{3} + \frac{31}{72} a^{2} - \frac{5}{72} a + \frac{11}{48}$, $\frac{1}{1296} a^{16} - \frac{1}{1296} a^{13} + \frac{1}{432} a^{12} - \frac{31}{432} a^{11} + \frac{11}{432} a^{10} - \frac{13}{432} a^{9} + \frac{13}{432} a^{8} - \frac{35}{1296} a^{7} - \frac{23}{432} a^{6} + \frac{173}{432} a^{5} + \frac{299}{1296} a^{4} - \frac{17}{48} a^{3} + \frac{1}{9} a^{2} - \frac{25}{144} a + \frac{5}{16}$, $\frac{1}{2592663696458124629647536} a^{17} + \frac{14127287952783247595}{1296331848229062314823768} a^{16} - \frac{58093014840062120731}{2592663696458124629647536} a^{15} - \frac{1675661703615518375347}{432110616076354104941256} a^{14} + \frac{5600689782801526096409}{1296331848229062314823768} a^{13} - \frac{10424854950172293707689}{2592663696458124629647536} a^{12} - \frac{17677395554496168803353}{324082962057265578705942} a^{11} + \frac{37320697189940813017421}{2592663696458124629647536} a^{10} - \frac{18049282320713283759671}{648165924114531157411884} a^{9} - \frac{3849616075826525067031}{288073744050902736627504} a^{8} - \frac{98308680095825821029155}{1296331848229062314823768} a^{7} + \frac{384723707089409724865381}{2592663696458124629647536} a^{6} + \frac{15600459441043260219805}{162041481028632789352971} a^{5} + \frac{19805991507636888050857}{96024581350300912209168} a^{4} + \frac{18798696838637778286291}{96024581350300912209168} a^{3} - \frac{92634297885391852362119}{288073744050902736627504} a^{2} + \frac{999265645369158351701}{32008193783433637403056} a - \frac{7759090429498184638493}{32008193783433637403056}$

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

Class group and class number

$C_{15}$, which has order $15$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 483473.437416 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-47}) \), 3.1.3807.1 x3, \(\Q(\zeta_{9})^+\), 6.0.681182703.1, 6.0.681182703.2, 6.0.8409663.1 x2, 9.3.55175798943.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.8409663.1
Degree 9 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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

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
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$47$47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$