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Properties

Label 18.0.28946065837...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{15}\cdot 7^{6}$
Root discriminant $20.11$
Ramified primes $2, 3, 5, 7$
Class number $2$
Class group $[2]$
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![9, 18, 30, 48, 25, 48, 1, -20, 21, -80, 50, -60, 50, -20, 15, -2, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 2*x^15 + 15*x^14 - 20*x^13 + 50*x^12 - 60*x^11 + 50*x^10 - 80*x^9 + 21*x^8 - 20*x^7 + x^6 + 48*x^5 + 25*x^4 + 48*x^3 + 30*x^2 + 18*x + 9)
 
gp: K = bnfinit(x^18 + x^16 - 2*x^15 + 15*x^14 - 20*x^13 + 50*x^12 - 60*x^11 + 50*x^10 - 80*x^9 + 21*x^8 - 20*x^7 + x^6 + 48*x^5 + 25*x^4 + 48*x^3 + 30*x^2 + 18*x + 9, 1)
 

Normalized defining polynomial

\( x^{18} + x^{16} - 2 x^{15} + 15 x^{14} - 20 x^{13} + 50 x^{12} - 60 x^{11} + 50 x^{10} - 80 x^{9} + 21 x^{8} - 20 x^{7} + x^{6} + 48 x^{5} + 25 x^{4} + 48 x^{3} + 30 x^{2} + 18 x + 9 \)

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:  \(-289460658375000000000000=-\,2^{12}\cdot 3^{9}\cdot 5^{15}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.11$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{13} + \frac{1}{18} a^{12} + \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{18} a^{9} + \frac{1}{9} a^{8} - \frac{2}{9} a^{7} + \frac{1}{18} a^{6} + \frac{2}{9} a^{5} - \frac{1}{9} a^{4} + \frac{5}{18} a^{3} + \frac{2}{9} a^{2} - \frac{1}{6}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{12} - \frac{1}{18} a^{11} + \frac{1}{18} a^{9} - \frac{1}{9} a^{8} - \frac{1}{6} a^{7} - \frac{2}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{2} a^{4} + \frac{7}{18} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{18} a^{16} - \frac{1}{18} a^{13} - \frac{1}{18} a^{12} + \frac{1}{18} a^{10} + \frac{1}{18} a^{9} + \frac{5}{18} a^{7} - \frac{2}{9} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{4}{9} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{50926662} a^{17} + \frac{486653}{25463331} a^{16} + \frac{474704}{25463331} a^{15} + \frac{110428}{25463331} a^{14} - \frac{1899805}{25463331} a^{13} - \frac{1093300}{25463331} a^{12} + \frac{931069}{16975554} a^{11} - \frac{578927}{25463331} a^{10} + \frac{2099803}{25463331} a^{9} - \frac{1329866}{8487777} a^{8} - \frac{24849905}{50926662} a^{7} - \frac{15808211}{50926662} a^{6} - \frac{8858815}{50926662} a^{5} + \frac{93821}{2995686} a^{4} - \frac{9524569}{25463331} a^{3} - \frac{4066798}{25463331} a^{2} - \frac{1233283}{8487777} a - \frac{3860143}{16975554}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23139.13431049142 \)
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

\(\Q(\sqrt{-15}) \), 3.1.300.1, 3.1.175.1, 6.0.1350000.1, 6.0.4134375.1, 9.1.9261000000.1

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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$7$7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$