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Properties

Label 18.18.4380468175...1648.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{27}\cdot 7^{12}\cdot 11^{9}$
Root discriminant $34.33$
Ramified primes $2, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![1, -18, -427, 1700, 2619, -11930, -423, 24924, -10133, -19102, 10757, 6348, -3992, -992, 632, 72, -43, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 43*x^16 + 72*x^15 + 632*x^14 - 992*x^13 - 3992*x^12 + 6348*x^11 + 10757*x^10 - 19102*x^9 - 10133*x^8 + 24924*x^7 - 423*x^6 - 11930*x^5 + 2619*x^4 + 1700*x^3 - 427*x^2 - 18*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 43*x^16 + 72*x^15 + 632*x^14 - 992*x^13 - 3992*x^12 + 6348*x^11 + 10757*x^10 - 19102*x^9 - 10133*x^8 + 24924*x^7 - 423*x^6 - 11930*x^5 + 2619*x^4 + 1700*x^3 - 427*x^2 - 18*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 43 x^{16} + 72 x^{15} + 632 x^{14} - 992 x^{13} - 3992 x^{12} + 6348 x^{11} + 10757 x^{10} - 19102 x^{9} - 10133 x^{8} + 24924 x^{7} - 423 x^{6} - 11930 x^{5} + 2619 x^{4} + 1700 x^{3} - 427 x^{2} - 18 x + 1 \)

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:  \(4380468175801074602525851648=2^{27}\cdot 7^{12}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.33$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{8916} a^{16} + \frac{4}{2229} a^{15} + \frac{49}{743} a^{14} - \frac{101}{4458} a^{13} - \frac{37}{743} a^{12} + \frac{329}{1486} a^{11} + \frac{275}{2229} a^{10} - \frac{189}{1486} a^{9} - \frac{883}{8916} a^{8} + \frac{631}{2229} a^{7} - \frac{15}{1486} a^{6} + \frac{845}{2229} a^{5} + \frac{933}{2972} a^{4} + \frac{781}{4458} a^{3} + \frac{1022}{2229} a^{2} - \frac{667}{2229} a + \frac{4445}{8916}$, $\frac{1}{23166039509494141668} a^{17} - \frac{217343734760233}{11583019754747070834} a^{16} - \frac{89922324609976163}{11583019754747070834} a^{15} + \frac{150009120514023827}{11583019754747070834} a^{14} + \frac{289239149694241435}{5791509877373535417} a^{13} + \frac{747720436271985869}{11583019754747070834} a^{12} - \frac{1958701998124930169}{11583019754747070834} a^{11} - \frac{2704038325849825991}{11583019754747070834} a^{10} + \frac{438261274095673723}{7722013169831380556} a^{9} - \frac{205370855351850185}{11583019754747070834} a^{8} + \frac{1135584883059831890}{5791509877373535417} a^{7} - \frac{163628028803595587}{1930503292457845139} a^{6} - \frac{3257620441035070013}{7722013169831380556} a^{5} + \frac{802653512947512571}{1930503292457845139} a^{4} - \frac{3290046270656853623}{11583019754747070834} a^{3} - \frac{1051161141256063577}{5791509877373535417} a^{2} + \frac{2089020871230126533}{7722013169831380556} a - \frac{1291526443201051621}{3861006584915690278}$

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:  \( 108234901.786 \) (assuming GRH)
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

\(\Q(\sqrt{22}) \), 3.3.4312.1 x3, \(\Q(\zeta_{7})^+\), 6.6.1636214272.1, 6.6.1636214272.2, 6.6.33392128.1 x2, 9.9.80174499328.1 x3

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

Sibling fields

Degree 6 sibling: 6.6.33392128.1
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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$