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Properties

Label 18.0.80269720285...2848.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{9}\cdot 7^{15}$
Root discriminant $31.24$
Ramified primes $2, 3, 7$
Class number $28$ (GRH)
Class group $[2, 14]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17851, -54474, 112877, -124430, 146017, -123990, 110878, -85698, 65630, -42600, 25573, -13076, 5951, -2306, 777, -214, 50, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 50*x^16 - 214*x^15 + 777*x^14 - 2306*x^13 + 5951*x^12 - 13076*x^11 + 25573*x^10 - 42600*x^9 + 65630*x^8 - 85698*x^7 + 110878*x^6 - 123990*x^5 + 146017*x^4 - 124430*x^3 + 112877*x^2 - 54474*x + 17851)
 
gp: K = bnfinit(x^18 - 8*x^17 + 50*x^16 - 214*x^15 + 777*x^14 - 2306*x^13 + 5951*x^12 - 13076*x^11 + 25573*x^10 - 42600*x^9 + 65630*x^8 - 85698*x^7 + 110878*x^6 - 123990*x^5 + 146017*x^4 - 124430*x^3 + 112877*x^2 - 54474*x + 17851, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 50 x^{16} - 214 x^{15} + 777 x^{14} - 2306 x^{13} + 5951 x^{12} - 13076 x^{11} + 25573 x^{10} - 42600 x^{9} + 65630 x^{8} - 85698 x^{7} + 110878 x^{6} - 123990 x^{5} + 146017 x^{4} - 124430 x^{3} + 112877 x^{2} - 54474 x + 17851 \)

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:  \(-802697202857257993500622848=-\,2^{33}\cdot 3^{9}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{26} a^{12} + \frac{3}{13} a^{11} - \frac{4}{13} a^{10} - \frac{4}{13} a^{9} - \frac{2}{13} a^{8} + \frac{6}{13} a^{7} + \frac{6}{13} a^{6} - \frac{1}{13} a^{5} - \frac{7}{26} a^{4} + \frac{4}{13} a^{3} - \frac{4}{13} a^{2} + \frac{2}{13} a - \frac{11}{26}$, $\frac{1}{26} a^{13} + \frac{4}{13} a^{11} - \frac{6}{13} a^{10} - \frac{4}{13} a^{9} + \frac{5}{13} a^{8} - \frac{4}{13} a^{7} + \frac{2}{13} a^{6} + \frac{5}{26} a^{5} - \frac{1}{13} a^{4} - \frac{2}{13} a^{3} - \frac{9}{26} a - \frac{6}{13}$, $\frac{1}{26} a^{14} - \frac{4}{13} a^{11} + \frac{2}{13} a^{10} - \frac{2}{13} a^{9} - \frac{1}{13} a^{8} + \frac{6}{13} a^{7} - \frac{1}{2} a^{6} - \frac{6}{13} a^{5} - \frac{6}{13} a^{3} + \frac{3}{26} a^{2} + \frac{4}{13} a + \frac{5}{13}$, $\frac{1}{26} a^{15} + \frac{5}{13} a^{10} + \frac{6}{13} a^{9} + \frac{3}{13} a^{8} + \frac{5}{26} a^{7} + \frac{3}{13} a^{6} + \frac{5}{13} a^{5} + \frac{5}{13} a^{4} - \frac{11}{26} a^{3} - \frac{2}{13} a^{2} - \frac{5}{13} a - \frac{5}{13}$, $\frac{1}{5122} a^{16} - \frac{8}{2561} a^{15} - \frac{5}{2561} a^{14} - \frac{81}{5122} a^{13} - \frac{22}{2561} a^{12} - \frac{1243}{2561} a^{11} - \frac{576}{2561} a^{10} - \frac{509}{2561} a^{9} + \frac{2467}{5122} a^{8} - \frac{1077}{2561} a^{7} + \frac{623}{2561} a^{6} - \frac{1777}{5122} a^{5} - \frac{77}{394} a^{4} + \frac{106}{2561} a^{3} + \frac{1267}{2561} a^{2} + \frac{2001}{5122} a + \frac{836}{2561}$, $\frac{1}{16948112406041061069013513586782} a^{17} - \frac{23179296699978200617536983}{1303700954310850851462577968214} a^{16} + \frac{31932066214729057400108910174}{8474056203020530534506756793391} a^{15} + \frac{58052679000513093115694138759}{16948112406041061069013513586782} a^{14} - \frac{18188469586368849042043741801}{16948112406041061069013513586782} a^{13} - \frac{69745030686169967928508588289}{16948112406041061069013513586782} a^{12} - \frac{2807492468291027542656281123184}{8474056203020530534506756793391} a^{11} + \frac{1627859383921640403757677063232}{8474056203020530534506756793391} a^{10} + \frac{4000553673830716723909281019471}{16948112406041061069013513586782} a^{9} + \frac{3192742643797820094168433750783}{16948112406041061069013513586782} a^{8} + \frac{3592422711488507675150019106772}{8474056203020530534506756793391} a^{7} + \frac{807325938184933631980752633775}{16948112406041061069013513586782} a^{6} + \frac{2064869573149156746592339386158}{8474056203020530534506756793391} a^{5} - \frac{2441075867639913460274894899233}{8474056203020530534506756793391} a^{4} + \frac{3697607281797195922916781007818}{8474056203020530534506756793391} a^{3} + \frac{8400734553191036795120557499327}{16948112406041061069013513586782} a^{2} + \frac{4784634957029955817112327887905}{16948112406041061069013513586782} a - \frac{3123587884354644645635076672263}{16948112406041061069013513586782}$

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

Class group and class number

$C_{2}\times C_{14}$, which has order $28$ (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:  \( -1 \) (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:  \( 53687.93588131197 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-42}) \), \(\Q(\zeta_{7})^+\), 3.1.1176.1, 6.0.929359872.1, 6.0.232339968.1, 9.3.1626379776.1

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 12 sibling: 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.12.24.315$x^{12} + 32 x^{11} - 10 x^{10} + 8 x^{9} - 18 x^{8} + 32 x^{7} + 20 x^{6} + 24 x^{5} - 24 x^{4} + 32 x^{3} + 16 x^{2} - 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed