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Properties

Label 18.18.3542920590...0208.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{27}\cdot 1129^{8}$
Root discriminant $64.31$
Ramified primes $2, 1129$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{18}$ (as 18T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, -340, -1052, 7002, 13790, -59508, 17204, 61321, -33384, -21375, 15816, 2218, -2972, 122, 232, -29, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 29*x^16 + 232*x^15 + 122*x^14 - 2972*x^13 + 2218*x^12 + 15816*x^11 - 21375*x^10 - 33384*x^9 + 61321*x^8 + 17204*x^7 - 59508*x^6 + 13790*x^5 + 7002*x^4 - 1052*x^3 - 340*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 - 29*x^16 + 232*x^15 + 122*x^14 - 2972*x^13 + 2218*x^12 + 15816*x^11 - 21375*x^10 - 33384*x^9 + 61321*x^8 + 17204*x^7 - 59508*x^6 + 13790*x^5 + 7002*x^4 - 1052*x^3 - 340*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 29 x^{16} + 232 x^{15} + 122 x^{14} - 2972 x^{13} + 2218 x^{12} + 15816 x^{11} - 21375 x^{10} - 33384 x^{9} + 61321 x^{8} + 17204 x^{7} - 59508 x^{6} + 13790 x^{5} + 7002 x^{4} - 1052 x^{3} - 340 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(354292059029529242987404048990208=2^{27}\cdot 1129^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.31$
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:  $2, 1129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{247} a^{16} + \frac{45}{247} a^{15} - \frac{6}{19} a^{14} - \frac{4}{19} a^{13} - \frac{8}{247} a^{12} - \frac{4}{19} a^{11} + \frac{40}{247} a^{10} - \frac{58}{247} a^{9} - \frac{27}{247} a^{8} - \frac{79}{247} a^{7} + \frac{44}{247} a^{6} + \frac{108}{247} a^{5} - \frac{44}{247} a^{4} - \frac{40}{247} a^{3} - \frac{88}{247} a^{2} + \frac{41}{247} a + \frac{49}{247}$, $\frac{1}{414472240267771870686359} a^{17} - \frac{785842384902962640590}{414472240267771870686359} a^{16} + \frac{9252774866019544700336}{37679294569797442789669} a^{15} - \frac{1166859356632057543680}{31882480020597836206643} a^{14} + \frac{16517097514594210389015}{37679294569797442789669} a^{13} - \frac{78882975309451152800908}{414472240267771870686359} a^{12} - \frac{3895358134078730627102}{21814328435145887930861} a^{11} + \frac{14669441250311918461715}{37679294569797442789669} a^{10} + \frac{7970878544796715832083}{21814328435145887930861} a^{9} + \frac{5041959967491226832121}{414472240267771870686359} a^{8} - \frac{102366164730786038396057}{414472240267771870686359} a^{7} - \frac{106170605968571494157218}{414472240267771870686359} a^{6} + \frac{127649211742329088340062}{414472240267771870686359} a^{5} + \frac{103807582824091100451443}{414472240267771870686359} a^{4} - \frac{23551715266790177602933}{414472240267771870686359} a^{3} - \frac{5964951779576962019084}{31882480020597836206643} a^{2} - \frac{1703386057391525924702}{31882480020597836206643} a - \frac{119872057065317971629720}{414472240267771870686359}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 26253722476.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{18}$ (as 18T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $D_{18}$
Character table for $D_{18}$

Intermediate fields

\(\Q(\sqrt{2}) \), 3.3.1129.1, 6.6.652616192.2, 9.9.1624709678881.1

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 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
1129Data not computed