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Properties

Label 18.0.88858361481...6323.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{8}\cdot 23^{8}$
Root discriminant $16.57$
Ramified primes $3, 7, 23$
Class number $1$
Class group Trivial
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -20, 56, -138, 207, -216, 171, -100, 56, -40, 20, 7, -26, 24, -8, -3, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 3*x^15 - 8*x^14 + 24*x^13 - 26*x^12 + 7*x^11 + 20*x^10 - 40*x^9 + 56*x^8 - 100*x^7 + 171*x^6 - 216*x^5 + 207*x^4 - 138*x^3 + 56*x^2 - 20*x + 25)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 3*x^15 - 8*x^14 + 24*x^13 - 26*x^12 + 7*x^11 + 20*x^10 - 40*x^9 + 56*x^8 - 100*x^7 + 171*x^6 - 216*x^5 + 207*x^4 - 138*x^3 + 56*x^2 - 20*x + 25, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} - 3 x^{15} - 8 x^{14} + 24 x^{13} - 26 x^{12} + 7 x^{11} + 20 x^{10} - 40 x^{9} + 56 x^{8} - 100 x^{7} + 171 x^{6} - 216 x^{5} + 207 x^{4} - 138 x^{3} + 56 x^{2} - 20 x + 25 \)

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:  \(-8885836148113812196323=-\,3^{9}\cdot 7^{8}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.57$
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:  $3, 7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1405} a^{16} + \frac{118}{281} a^{15} - \frac{649}{1405} a^{14} - \frac{133}{281} a^{13} + \frac{602}{1405} a^{12} - \frac{14}{281} a^{11} + \frac{109}{1405} a^{10} - \frac{156}{1405} a^{9} + \frac{587}{1405} a^{8} + \frac{316}{1405} a^{7} + \frac{439}{1405} a^{6} + \frac{387}{1405} a^{5} - \frac{13}{1405} a^{4} + \frac{16}{281} a^{3} + \frac{287}{1405} a^{2} + \frac{443}{1405} a + \frac{31}{281}$, $\frac{1}{1296084581245} a^{17} - \frac{330196483}{1296084581245} a^{16} - \frac{234895413309}{1296084581245} a^{15} + \frac{257358394507}{1296084581245} a^{14} + \frac{221897008377}{1296084581245} a^{13} + \frac{310683828874}{1296084581245} a^{12} - \frac{727301096}{4612400645} a^{11} - \frac{441734178503}{1296084581245} a^{10} - \frac{43499383011}{259216916249} a^{9} + \frac{108629732593}{259216916249} a^{8} - \frac{4454372507}{76240269485} a^{7} + \frac{89292854209}{259216916249} a^{6} - \frac{235090669464}{1296084581245} a^{5} - \frac{30793027721}{1296084581245} a^{4} - \frac{138333642613}{1296084581245} a^{3} - \frac{639520136738}{1296084581245} a^{2} + \frac{590754703921}{1296084581245} a + \frac{11569082163}{259216916249}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{86523088}{4612400645} a^{17} - \frac{179592228}{4612400645} a^{16} + \frac{202658678}{4612400645} a^{15} + \frac{335823602}{4612400645} a^{14} - \frac{1052114249}{4612400645} a^{13} + \frac{1118460544}{4612400645} a^{12} + \frac{191419542}{4612400645} a^{11} - \frac{428541938}{922480129} a^{10} + \frac{1964417304}{4612400645} a^{9} - \frac{1244774728}{4612400645} a^{8} + \frac{26407737}{271317685} a^{7} - \frac{3299317316}{4612400645} a^{6} + \frac{962275944}{922480129} a^{5} - \frac{2573579506}{4612400645} a^{4} - \frac{2656532934}{4612400645} a^{3} + \frac{6750421748}{4612400645} a^{2} - \frac{3811344654}{4612400645} a + \frac{560767290}{922480129} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9914.42923149 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.23.1, 6.0.14283.1, 9.1.671898241.1

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

Sibling fields

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 $18$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
3Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$