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Properties

Label 18.18.4310084312...0112.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 7^{15}\cdot 53^{6}$
Root discriminant $30.18$
Ramified primes $2, 7, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6\times S_4$ (as 18T61)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 35, -59, -464, 961, 1433, -4279, -110, 6303, -2725, -3688, 2639, 729, -898, 28, 116, -17, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 17*x^16 + 116*x^15 + 28*x^14 - 898*x^13 + 729*x^12 + 2639*x^11 - 3688*x^10 - 2725*x^9 + 6303*x^8 - 110*x^7 - 4279*x^6 + 1433*x^5 + 961*x^4 - 464*x^3 - 59*x^2 + 35*x + 1)
 
gp: K = bnfinit(x^18 - 5*x^17 - 17*x^16 + 116*x^15 + 28*x^14 - 898*x^13 + 729*x^12 + 2639*x^11 - 3688*x^10 - 2725*x^9 + 6303*x^8 - 110*x^7 - 4279*x^6 + 1433*x^5 + 961*x^4 - 464*x^3 - 59*x^2 + 35*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 17 x^{16} + 116 x^{15} + 28 x^{14} - 898 x^{13} + 729 x^{12} + 2639 x^{11} - 3688 x^{10} - 2725 x^{9} + 6303 x^{8} - 110 x^{7} - 4279 x^{6} + 1433 x^{5} + 961 x^{4} - 464 x^{3} - 59 x^{2} + 35 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(431008431261766353738330112=2^{12}\cdot 7^{15}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.18$
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:  $2, 7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{138590226618196} a^{17} - \frac{245490949241}{138590226618196} a^{16} - \frac{4275911004508}{34647556654549} a^{15} - \frac{4904363186689}{34647556654549} a^{14} - \frac{10801836929197}{34647556654549} a^{13} - \frac{30784874560885}{69295113309098} a^{12} - \frac{63336249467859}{138590226618196} a^{11} - \frac{58187526492763}{138590226618196} a^{10} + \frac{13571099309751}{138590226618196} a^{9} - \frac{9395538380189}{138590226618196} a^{8} + \frac{6626329174027}{69295113309098} a^{7} - \frac{3493320983698}{34647556654549} a^{6} - \frac{30685660747647}{138590226618196} a^{5} - \frac{35007708781051}{138590226618196} a^{4} + \frac{4704834458119}{69295113309098} a^{3} - \frac{23902785662391}{69295113309098} a^{2} + \frac{63962116244769}{138590226618196} a - \frac{33682773614051}{138590226618196}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 23947078.6456 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_4$ (as 18T61):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 30 conjugacy class representatives for $C_6\times S_4$
Character table for $C_6\times S_4$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 6.6.755373808.1, 9.9.17515230173.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.18$x^{12} + 80 x^{10} + 81 x^{8} - 160 x^{6} - 117 x^{4} + 80 x^{2} + 227$$2$$6$$12$$D_4 \times C_3$$[2, 2]^{6}$
7Data not computed
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$