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Properties

Label 18.0.42845606719...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{21}\cdot 5^{6}$
Root discriminant $12.32$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_2\times S_3^2$ (as 18T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 24, -78, 210, -471, 896, -1455, 2019, -2374, 2358, -1974, 1393, -822, 399, -154, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 154*x^15 + 399*x^14 - 822*x^13 + 1393*x^12 - 1974*x^11 + 2358*x^10 - 2374*x^9 + 2019*x^8 - 1455*x^7 + 896*x^6 - 471*x^5 + 210*x^4 - 78*x^3 + 24*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 154*x^15 + 399*x^14 - 822*x^13 + 1393*x^12 - 1974*x^11 + 2358*x^10 - 2374*x^9 + 2019*x^8 - 1455*x^7 + 896*x^6 - 471*x^5 + 210*x^4 - 78*x^3 + 24*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 154 x^{15} + 399 x^{14} - 822 x^{13} + 1393 x^{12} - 1974 x^{11} + 2358 x^{10} - 2374 x^{9} + 2019 x^{8} - 1455 x^{7} + 896 x^{6} - 471 x^{5} + 210 x^{4} - 78 x^{3} + 24 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-42845606719488000000=-\,2^{18}\cdot 3^{21}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.32$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} + \frac{1}{13} a^{15} - \frac{1}{13} a^{14} + \frac{1}{13} a^{13} - \frac{3}{13} a^{12} + \frac{2}{13} a^{11} - \frac{5}{13} a^{10} - \frac{4}{13} a^{9} - \frac{2}{13} a^{8} + \frac{1}{13} a^{7} - \frac{4}{13} a^{6} - \frac{4}{13} a^{5} + \frac{1}{13} a^{4} - \frac{3}{13} a^{3} - \frac{6}{13} a^{2} + \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{13949} a^{17} + \frac{8}{377} a^{16} - \frac{6245}{13949} a^{15} - \frac{6742}{13949} a^{14} - \frac{6481}{13949} a^{13} + \frac{4304}{13949} a^{12} - \frac{24}{1073} a^{11} + \frac{2655}{13949} a^{10} - \frac{2274}{13949} a^{9} - \frac{2786}{13949} a^{8} + \frac{1032}{13949} a^{7} - \frac{6449}{13949} a^{6} - \frac{3532}{13949} a^{5} + \frac{5999}{13949} a^{4} + \frac{3659}{13949} a^{3} - \frac{3222}{13949} a^{2} + \frac{1255}{13949} a - \frac{3511}{13949}$

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:  \( -\frac{227}{377} a^{17} + \frac{1915}{377} a^{16} - \frac{9214}{377} a^{15} + \frac{30235}{377} a^{14} - \frac{75151}{377} a^{13} + \frac{147612}{377} a^{12} - \frac{237330}{377} a^{11} + \frac{24297}{29} a^{10} - \frac{350612}{377} a^{9} + \frac{24763}{29} a^{8} - \frac{245835}{377} a^{7} + \frac{157154}{377} a^{6} - \frac{88797}{377} a^{5} + \frac{44176}{377} a^{4} - \frac{19637}{377} a^{3} + \frac{6858}{377} a^{2} - \frac{2425}{377} a + \frac{802}{377} \) (order $6$)
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:  \( 735.644505219 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.216.1, 3.1.135.1, 6.0.139968.1, 6.0.54675.1, 9.1.3779136000.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: 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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.11$x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$