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Properties

Label 18.0.19110297600...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{6}\cdot 5^{15}$
Root discriminant $19.65$
Ramified primes $2, 3, 5$
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![1, 0, -6, 14, 20, -100, 155, -20, -70, 20, 44, 20, -19, -24, 0, 14, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 14*x^15 - 24*x^13 - 19*x^12 + 20*x^11 + 44*x^10 + 20*x^9 - 70*x^8 - 20*x^7 + 155*x^6 - 100*x^5 + 20*x^4 + 14*x^3 - 6*x^2 + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 14*x^15 - 24*x^13 - 19*x^12 + 20*x^11 + 44*x^10 + 20*x^9 - 70*x^8 - 20*x^7 + 155*x^6 - 100*x^5 + 20*x^4 + 14*x^3 - 6*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 14 x^{15} - 24 x^{13} - 19 x^{12} + 20 x^{11} + 44 x^{10} + 20 x^{9} - 70 x^{8} - 20 x^{7} + 155 x^{6} - 100 x^{5} + 20 x^{4} + 14 x^{3} - 6 x^{2} + 1 \)

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:  \(-191102976000000000000000=-\,2^{33}\cdot 3^{6}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.65$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23} a^{15} - \frac{3}{23} a^{13} + \frac{9}{23} a^{12} - \frac{1}{23} a^{11} - \frac{7}{23} a^{10} - \frac{4}{23} a^{9} - \frac{5}{23} a^{8} + \frac{10}{23} a^{7} + \frac{1}{23} a^{6} + \frac{7}{23} a^{5} + \frac{6}{23} a^{4} + \frac{4}{23} a^{3} - \frac{7}{23} a^{2} - \frac{1}{23} a - \frac{10}{23}$, $\frac{1}{23} a^{16} - \frac{3}{23} a^{14} + \frac{9}{23} a^{13} - \frac{1}{23} a^{12} - \frac{7}{23} a^{11} - \frac{4}{23} a^{10} - \frac{5}{23} a^{9} + \frac{10}{23} a^{8} + \frac{1}{23} a^{7} + \frac{7}{23} a^{6} + \frac{6}{23} a^{5} + \frac{4}{23} a^{4} - \frac{7}{23} a^{3} - \frac{1}{23} a^{2} - \frac{10}{23} a$, $\frac{1}{1111430110003} a^{17} - \frac{13638665708}{1111430110003} a^{16} - \frac{18586181325}{1111430110003} a^{15} + \frac{226495970633}{1111430110003} a^{14} + \frac{503913187900}{1111430110003} a^{13} - \frac{321247456365}{1111430110003} a^{12} + \frac{471921347278}{1111430110003} a^{11} - \frac{154668505774}{1111430110003} a^{10} + \frac{195948929056}{1111430110003} a^{9} - \frac{273164059110}{1111430110003} a^{8} + \frac{183209279501}{1111430110003} a^{7} + \frac{183935905997}{1111430110003} a^{6} - \frac{420815738415}{1111430110003} a^{5} - \frac{235668509668}{1111430110003} a^{4} + \frac{555539801052}{1111430110003} a^{3} - \frac{8842860716}{38325176207} a^{2} + \frac{374180109884}{1111430110003} a - \frac{287305224756}{1111430110003}$

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:  \( 18590.202593393617 \)
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

\(\Q(\sqrt{-10}) \), 3.1.200.1, 3.1.300.1, 6.0.1600000.1, 6.0.57600000.1, 9.1.3456000000.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.6.11.13$x^{6} + 10$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.12.22.79$x^{12} + 2 x^{10} + 4 x^{8} + 4 x^{6} + 4 x^{4} + 4$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$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}$