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Properties

Label 18.0.75874738678...9367.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{13}\cdot 23^{8}$
Root discriminant $16.43$
Ramified primes $7, 23$
Class number $1$
Class group Trivial
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -6, 9, 3, -23, 98, -186, 368, -418, 504, -399, 330, -189, 112, -44, 18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 18*x^16 - 44*x^15 + 112*x^14 - 189*x^13 + 330*x^12 - 399*x^11 + 504*x^10 - 418*x^9 + 368*x^8 - 186*x^7 + 98*x^6 - 23*x^5 + 3*x^4 + 9*x^3 - 6*x^2 + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 18*x^16 - 44*x^15 + 112*x^14 - 189*x^13 + 330*x^12 - 399*x^11 + 504*x^10 - 418*x^9 + 368*x^8 - 186*x^7 + 98*x^6 - 23*x^5 + 3*x^4 + 9*x^3 - 6*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 18 x^{16} - 44 x^{15} + 112 x^{14} - 189 x^{13} + 330 x^{12} - 399 x^{11} + 504 x^{10} - 418 x^{9} + 368 x^{8} - 186 x^{7} + 98 x^{6} - 23 x^{5} + 3 x^{4} + 9 x^{3} - 6 x^{2} + 1 \)

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:  \(-7587473867873232819367=-\,7^{13}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.43$
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:  $7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{7} a^{6} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{15} - \frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{112095305} a^{17} + \frac{712121}{16013615} a^{16} - \frac{1436612}{22419061} a^{15} + \frac{1050963}{16013615} a^{14} + \frac{6468498}{112095305} a^{13} - \frac{4004711}{112095305} a^{12} + \frac{728367}{16013615} a^{11} - \frac{9993980}{22419061} a^{10} - \frac{36073951}{112095305} a^{9} + \frac{1550666}{112095305} a^{8} - \frac{29081056}{112095305} a^{7} + \frac{39998083}{112095305} a^{6} + \frac{4560706}{112095305} a^{5} + \frac{6730973}{112095305} a^{4} + \frac{34375361}{112095305} a^{3} - \frac{4135983}{22419061} a^{2} + \frac{22464784}{112095305} a - \frac{9048191}{112095305}$

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:  $8$
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:  \( 3208.93860661 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.23.1, 6.0.181447.1, 9.1.671898241.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$