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Properties

Label 18.0.10968475320...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{21}\cdot 5^{6}$
Root discriminant $16.77$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -96, 96, 168, -144, -114, 93, 132, -81, -146, 42, 12, 75, -18, 6, -12, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 12*x^15 + 6*x^14 - 18*x^13 + 75*x^12 + 12*x^11 + 42*x^10 - 146*x^9 - 81*x^8 + 132*x^7 + 93*x^6 - 114*x^5 - 144*x^4 + 168*x^3 + 96*x^2 - 96*x + 64)
 
gp: K = bnfinit(x^18 + 3*x^16 - 12*x^15 + 6*x^14 - 18*x^13 + 75*x^12 + 12*x^11 + 42*x^10 - 146*x^9 - 81*x^8 + 132*x^7 + 93*x^6 - 114*x^5 - 144*x^4 + 168*x^3 + 96*x^2 - 96*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} - 12 x^{15} + 6 x^{14} - 18 x^{13} + 75 x^{12} + 12 x^{11} + 42 x^{10} - 146 x^{9} - 81 x^{8} + 132 x^{7} + 93 x^{6} - 114 x^{5} - 144 x^{4} + 168 x^{3} + 96 x^{2} - 96 x + 64 \)

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:  \(-10968475320188928000000=-\,2^{26}\cdot 3^{21}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.77$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{18} a^{9} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{2}{9}$, $\frac{1}{18} a^{10} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{18} a^{11} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{5}{18} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{18} a^{12} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{18} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{18} a^{13} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{108} a^{14} - \frac{1}{54} a^{13} + \frac{1}{108} a^{12} + \frac{1}{54} a^{11} + \frac{1}{54} a^{10} + \frac{1}{54} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{6} + \frac{11}{27} a^{5} - \frac{7}{108} a^{4} + \frac{11}{27} a^{3} + \frac{43}{108} a^{2} - \frac{5}{27} a + \frac{13}{27}$, $\frac{1}{648} a^{15} + \frac{1}{216} a^{13} + \frac{1}{162} a^{12} + \frac{1}{108} a^{11} - \frac{1}{108} a^{10} - \frac{5}{648} a^{9} - \frac{1}{36} a^{7} + \frac{13}{324} a^{6} + \frac{1}{24} a^{5} + \frac{23}{54} a^{4} - \frac{31}{648} a^{3} + \frac{17}{108} a^{2} - \frac{5}{27} a + \frac{31}{81}$, $\frac{1}{3888} a^{16} + \frac{1}{1944} a^{15} + \frac{1}{1296} a^{14} + \frac{5}{1944} a^{13} + \frac{7}{1944} a^{12} + \frac{7}{648} a^{11} - \frac{17}{3888} a^{10} - \frac{23}{1944} a^{9} + \frac{5}{216} a^{8} - \frac{59}{1944} a^{7} - \frac{29}{3888} a^{6} - \frac{89}{648} a^{5} - \frac{1531}{3888} a^{4} + \frac{97}{243} a^{3} - \frac{95}{324} a^{2} + \frac{109}{486} a - \frac{95}{243}$, $\frac{1}{2185056} a^{17} - \frac{31}{273132} a^{16} + \frac{811}{2185056} a^{15} - \frac{1391}{546264} a^{14} - \frac{8065}{1092528} a^{13} - \frac{12973}{1092528} a^{12} + \frac{54859}{2185056} a^{11} - \frac{12395}{546264} a^{10} + \frac{7529}{1092528} a^{9} + \frac{71635}{1092528} a^{8} + \frac{158207}{2185056} a^{7} + \frac{23951}{546264} a^{6} + \frac{80741}{2185056} a^{5} - \frac{27449}{1092528} a^{4} + \frac{35209}{273132} a^{3} - \frac{18133}{136566} a^{2} + \frac{15388}{68283} a + \frac{2563}{68283}$

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:  \( -\frac{527}{68283} a^{17} + \frac{3025}{1092528} a^{16} - \frac{10927}{546264} a^{15} + \frac{116503}{1092528} a^{14} - \frac{36979}{546264} a^{13} + \frac{73997}{546264} a^{12} - \frac{365083}{546264} a^{11} + \frac{76399}{1092528} a^{10} - \frac{78283}{546264} a^{9} + \frac{907951}{546264} a^{8} + \frac{307129}{546264} a^{7} - \frac{1499101}{1092528} a^{6} - \frac{736595}{546264} a^{5} + \frac{1194737}{1092528} a^{4} + \frac{215359}{136566} a^{3} - \frac{61522}{68283} a^{2} - \frac{84313}{68283} a + \frac{78263}{68283} \) (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:  \( 38983.74877017534 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 3.1.1080.1, 6.0.3499200.1, 6.0.34992.1, 9.1.60466176000.1 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$