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Properties

Label 18.6.81531076016...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{30}\cdot 3^{32}\cdot 5^{6}\cdot 11^{6}\cdot 23^{6}$
Root discriminant $242.08$
Ramified primes $2, 3, 5, 11, 23$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 18T269

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-262254607552729, 0, -40426599583227, 0, 503949705963, 0, 176355676530, 0, 3421153032, 0, -149831913, 0, -6124047, 0, -68853, 0, -138, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 138*x^16 - 68853*x^14 - 6124047*x^12 - 149831913*x^10 + 3421153032*x^8 + 176355676530*x^6 + 503949705963*x^4 - 40426599583227*x^2 - 262254607552729)
 
gp: K = bnfinit(x^18 - 138*x^16 - 68853*x^14 - 6124047*x^12 - 149831913*x^10 + 3421153032*x^8 + 176355676530*x^6 + 503949705963*x^4 - 40426599583227*x^2 - 262254607552729, 1)
 

Normalized defining polynomial

\( x^{18} - 138 x^{16} - 68853 x^{14} - 6124047 x^{12} - 149831913 x^{10} + 3421153032 x^{8} + 176355676530 x^{6} + 503949705963 x^{4} - 40426599583227 x^{2} - 262254607552729 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8153107601695937311094624312805556224000000=2^{30}\cdot 3^{32}\cdot 5^{6}\cdot 11^{6}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $242.08$
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:  $2, 3, 5, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{253} a^{8} + \frac{5}{11} a^{6} - \frac{37}{253} a^{4} + \frac{71}{253} a^{2}$, $\frac{1}{253} a^{9} + \frac{5}{11} a^{7} - \frac{37}{253} a^{5} + \frac{71}{253} a^{3}$, $\frac{1}{64009} a^{10} + \frac{5}{2783} a^{8} + \frac{24251}{64009} a^{6} + \frac{11456}{64009} a^{4} + \frac{123}{253} a^{2}$, $\frac{1}{64009} a^{11} + \frac{5}{2783} a^{9} + \frac{24251}{64009} a^{7} + \frac{11456}{64009} a^{5} + \frac{123}{253} a^{3}$, $\frac{1}{16194277} a^{12} + \frac{5}{704099} a^{10} + \frac{24251}{16194277} a^{8} + \frac{7372491}{16194277} a^{6} - \frac{14045}{64009} a^{4} + \frac{6}{253} a^{2}$, $\frac{1}{16194277} a^{13} + \frac{5}{704099} a^{11} + \frac{24251}{16194277} a^{9} + \frac{7372491}{16194277} a^{7} - \frac{14045}{64009} a^{5} + \frac{6}{253} a^{3}$, $\frac{1}{4097152081} a^{14} + \frac{5}{178137047} a^{12} + \frac{24251}{4097152081} a^{10} + \frac{7372491}{4097152081} a^{8} - \frac{3278504}{16194277} a^{6} - \frac{13150}{64009} a^{4} - \frac{113}{253} a^{2}$, $\frac{1}{4097152081} a^{15} + \frac{5}{178137047} a^{13} + \frac{24251}{4097152081} a^{11} + \frac{7372491}{4097152081} a^{9} - \frac{3278504}{16194277} a^{7} - \frac{13150}{64009} a^{5} - \frac{113}{253} a^{3}$, $\frac{1}{137778905817168744772458436084287121} a^{16} - \frac{243323262951387621270934}{5990387209442119337932975481925527} a^{14} - \frac{3294749068712987899865774531}{137778905817168744772458436084287121} a^{12} - \frac{5893802546959349386667671440}{137778905817168744772458436084287121} a^{10} - \frac{901884679889145924954748920061}{544580655403829030721179589265957} a^{8} + \frac{484459264517100784882961317469}{2152492709106043599688456874569} a^{6} - \frac{2634511156355108577336215520}{8507876320577247429598643773} a^{4} - \frac{3113806364568274834376550}{33627969646550385097227841} a^{2} + \frac{64659282707264427291557}{132916876073321680226197}$, $\frac{1}{137778905817168744772458436084287121} a^{17} - \frac{243323262951387621270934}{5990387209442119337932975481925527} a^{15} - \frac{3294749068712987899865774531}{137778905817168744772458436084287121} a^{13} - \frac{5893802546959349386667671440}{137778905817168744772458436084287121} a^{11} - \frac{901884679889145924954748920061}{544580655403829030721179589265957} a^{9} + \frac{484459264517100784882961317469}{2152492709106043599688456874569} a^{7} - \frac{2634511156355108577336215520}{8507876320577247429598643773} a^{5} - \frac{3113806364568274834376550}{33627969646550385097227841} a^{3} + \frac{64659282707264427291557}{132916876073321680226197} a$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $11$
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:  \( 171435305330000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T269:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 24 conjugacy class representatives for t18n269
Character table for t18n269 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.1620.1, 9.9.344373768000.1

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

Sibling fields

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 R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$3$3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
$23$23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.12.6.2$x^{12} - 6436343 x^{2} + 2220538335$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$