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Properties

Label 18.12.1089427651...0464.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{33}\cdot 37^{6}\cdot 367^{3}$
Root discriminant $31.77$
Ramified primes $2, 37, 367$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times S_3\wr C_2$ (as 18T150)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 0, -160, 224, 600, -1168, -904, 2400, 736, -2424, -344, 1376, 30, -464, 60, 72, -16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 16*x^16 + 72*x^15 + 60*x^14 - 464*x^13 + 30*x^12 + 1376*x^11 - 344*x^10 - 2424*x^9 + 736*x^8 + 2400*x^7 - 904*x^6 - 1168*x^5 + 600*x^4 + 224*x^3 - 160*x^2 + 8)
 
gp: K = bnfinit(x^18 - 4*x^17 - 16*x^16 + 72*x^15 + 60*x^14 - 464*x^13 + 30*x^12 + 1376*x^11 - 344*x^10 - 2424*x^9 + 736*x^8 + 2400*x^7 - 904*x^6 - 1168*x^5 + 600*x^4 + 224*x^3 - 160*x^2 + 8, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 16 x^{16} + 72 x^{15} + 60 x^{14} - 464 x^{13} + 30 x^{12} + 1376 x^{11} - 344 x^{10} - 2424 x^{9} + 736 x^{8} + 2400 x^{7} - 904 x^{6} - 1168 x^{5} + 600 x^{4} + 224 x^{3} - 160 x^{2} + 8 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1089427651175529674612670464=-\,2^{33}\cdot 37^{6}\cdot 367^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.77$
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:  $2, 37, 367$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{4} a^{14}$, $\frac{1}{4} a^{15}$, $\frac{1}{4} a^{16}$, $\frac{1}{12320231540324} a^{17} + \frac{729746468155}{12320231540324} a^{16} + \frac{15245782387}{424835570356} a^{15} + \frac{525317980929}{12320231540324} a^{14} - \frac{763818227999}{6160115770162} a^{13} - \frac{319283710221}{3080057885081} a^{12} + \frac{347998214531}{3080057885081} a^{11} + \frac{1568537066}{31753174073} a^{10} - \frac{352629019915}{6160115770162} a^{9} - \frac{988147003397}{6160115770162} a^{8} + \frac{109215843431}{3080057885081} a^{7} + \frac{1277693900281}{6160115770162} a^{6} - \frac{738266602786}{3080057885081} a^{5} - \frac{843731613099}{3080057885081} a^{4} - \frac{844154837243}{3080057885081} a^{3} - \frac{1183063803657}{3080057885081} a^{2} - \frac{126562163355}{3080057885081} a - \frac{164482289888}{3080057885081}$

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:  $14$
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:  \( 13339016.9312 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_3\wr C_2$ (as 18T150):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 27 conjugacy class representatives for $S_3\times S_3\wr C_2$
Character table for $S_3\times S_3\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 3.3.148.1, 6.6.2803712.1, 6.4.187904.1

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 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$37$37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$
37.12.6.1$x^{12} + 2026120 x^{6} - 69343957 x^{2} + 1026290563600$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
367Data not computed