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Properties

Label 18.6.18840693011...5989.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{21}\cdot 23^{9}$
Root discriminant $17.28$
Ramified primes $3, 23$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, -72, 214, -186, -252, 481, 120, -528, -28, 384, -12, -185, 12, 60, -3, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^16 - 3*x^15 + 60*x^14 + 12*x^13 - 185*x^12 - 12*x^11 + 384*x^10 - 28*x^9 - 528*x^8 + 120*x^7 + 481*x^6 - 252*x^5 - 186*x^4 + 214*x^3 - 72*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^18 - 12*x^16 - 3*x^15 + 60*x^14 + 12*x^13 - 185*x^12 - 12*x^11 + 384*x^10 - 28*x^9 - 528*x^8 + 120*x^7 + 481*x^6 - 252*x^5 - 186*x^4 + 214*x^3 - 72*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{16} - 3 x^{15} + 60 x^{14} + 12 x^{13} - 185 x^{12} - 12 x^{11} + 384 x^{10} - 28 x^{9} - 528 x^{8} + 120 x^{7} + 481 x^{6} - 252 x^{5} - 186 x^{4} + 214 x^{3} - 72 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18840693011426466715989=3^{21}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.28$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{14} + \frac{3}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{49} a^{16} - \frac{2}{49} a^{15} - \frac{17}{49} a^{14} + \frac{1}{7} a^{13} + \frac{10}{49} a^{12} - \frac{22}{49} a^{11} + \frac{2}{7} a^{10} - \frac{17}{49} a^{9} + \frac{5}{49} a^{8} - \frac{11}{49} a^{7} - \frac{19}{49} a^{6} - \frac{23}{49} a^{5} - \frac{23}{49} a^{4} + \frac{8}{49} a^{3} + \frac{12}{49} a^{2} - \frac{22}{49} a - \frac{10}{49}$, $\frac{1}{443042028401} a^{17} - \frac{1230412426}{443042028401} a^{16} + \frac{18142907100}{443042028401} a^{15} - \frac{136516108020}{443042028401} a^{14} - \frac{149354551392}{443042028401} a^{13} + \frac{119440472743}{443042028401} a^{12} - \frac{32756426007}{443042028401} a^{11} + \frac{55287958471}{443042028401} a^{10} + \frac{7467887208}{63291718343} a^{9} + \frac{130423486363}{443042028401} a^{8} + \frac{206656416}{63291718343} a^{7} - \frac{179731248285}{443042028401} a^{6} + \frac{13287471873}{443042028401} a^{5} - \frac{1287386704}{9041674049} a^{4} - \frac{151571452919}{443042028401} a^{3} - \frac{51923426872}{443042028401} a^{2} + \frac{5197227613}{63291718343} a - \frac{4066126937}{443042028401}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  \( 12848.0715901 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{69}) \), 3.3.621.1 x3, 3.1.23.1, 6.2.328509.1, 6.6.26609229.1, 9.3.5508110403.1 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: 6.0.1156923.2
Degree 9 sibling: 9.3.5508110403.1
Degree 12 sibling: 12.0.708051067974441.1
Degree 18 siblings: 18.0.819160565714194205043.1, 18.0.697803444867646915407.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$