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Properties

Label 18.0.48036106180...0128.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 7^{12}$
Root discriminant $26.71$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -26, 220, -730, 1588, -2148, 839, 1954, -3170, 1736, 298, -1110, 923, -488, 202, -70, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 22*x^16 - 70*x^15 + 202*x^14 - 488*x^13 + 923*x^12 - 1110*x^11 + 298*x^10 + 1736*x^9 - 3170*x^8 + 1954*x^7 + 839*x^6 - 2148*x^5 + 1588*x^4 - 730*x^3 + 220*x^2 - 26*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 22*x^16 - 70*x^15 + 202*x^14 - 488*x^13 + 923*x^12 - 1110*x^11 + 298*x^10 + 1736*x^9 - 3170*x^8 + 1954*x^7 + 839*x^6 - 2148*x^5 + 1588*x^4 - 730*x^3 + 220*x^2 - 26*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 22 x^{16} - 70 x^{15} + 202 x^{14} - 488 x^{13} + 923 x^{12} - 1110 x^{11} + 298 x^{10} + 1736 x^{9} - 3170 x^{8} + 1954 x^{7} + 839 x^{6} - 2148 x^{5} + 1588 x^{4} - 730 x^{3} + 220 x^{2} - 26 x + 1 \)

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:  \(-48036106180347836985520128=-\,2^{12}\cdot 3^{25}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.71$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{15} - \frac{1}{6} a^{13} - \frac{1}{4} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{4} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{12}$, $\frac{1}{1956} a^{16} - \frac{27}{652} a^{15} - \frac{50}{489} a^{14} + \frac{151}{652} a^{13} + \frac{193}{1956} a^{12} - \frac{167}{978} a^{11} - \frac{25}{489} a^{10} - \frac{71}{489} a^{9} - \frac{117}{326} a^{8} - \frac{235}{978} a^{7} - \frac{293}{978} a^{6} - \frac{439}{978} a^{5} - \frac{31}{652} a^{4} - \frac{583}{1956} a^{3} - \frac{101}{978} a^{2} - \frac{511}{1956} a + \frac{115}{652}$, $\frac{1}{226347015203113212} a^{17} + \frac{27210532652383}{226347015203113212} a^{16} - \frac{485080589448461}{113173507601556606} a^{15} - \frac{965570774768317}{113173507601556606} a^{14} - \frac{8950106133134465}{113173507601556606} a^{13} - \frac{14598118421121505}{75449005067704404} a^{12} + \frac{5525055334268275}{113173507601556606} a^{11} + \frac{16159040290766605}{113173507601556606} a^{10} - \frac{3437391971168797}{37724502533852202} a^{9} + \frac{8857120307196962}{56586753800778303} a^{8} + \frac{822076957295306}{18862251266926101} a^{7} - \frac{32836114320071209}{113173507601556606} a^{6} - \frac{7000777817618941}{25149668355901468} a^{5} + \frac{7105398940536257}{25149668355901468} a^{4} - \frac{41125169768859565}{113173507601556606} a^{3} + \frac{2965722545607368}{18862251266926101} a^{2} - \frac{22070318956147826}{56586753800778303} a + \frac{39892088193243191}{226347015203113212}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{139063139641}{61330382934} a^{17} + \frac{547091680527}{40886921956} a^{16} - \frac{2979488258335}{61330382934} a^{15} + \frac{6297303461945}{40886921956} a^{14} - \frac{27177325773031}{61330382934} a^{13} + \frac{130478727022517}{122660765868} a^{12} - \frac{61041387893521}{30665191467} a^{11} + \frac{71345039720806}{30665191467} a^{10} - \frac{9336422718013}{20443460978} a^{9} - \frac{121805050491734}{30665191467} a^{8} + \frac{417109291292785}{61330382934} a^{7} - \frac{232328248160653}{61330382934} a^{6} - \frac{22964437977312}{10221730489} a^{5} + \frac{570093480885209}{122660765868} a^{4} - \frac{194126931977957}{61330382934} a^{3} + \frac{166872942983111}{122660765868} a^{2} - \frac{7592043247467}{20443460978} a + \frac{1027349423205}{40886921956} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 732150.8656444304 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.588.1 x3, 6.0.1037232.1, 9.3.1333834713792.1

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.11.11$x^{6} + 9 x^{3} + 21$$6$$1$$11$$S_3\times C_3$$[5/2]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.11.11$x^{6} + 9 x^{3} + 21$$6$$1$$11$$S_3\times C_3$$[5/2]_{2}^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$