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Properties

Label 18.2.47833862195...2272.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{44}\cdot 17^{9}$
Root discriminant $95.99$
Ramified primes $2, 3, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-512704, 1155456, -3052656, 1788924, 520524, 139896, -793350, 232506, 204984, -76955, -35847, 16326, 4032, -2304, -216, 192, 0, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 192*x^15 - 216*x^14 - 2304*x^13 + 4032*x^12 + 16326*x^11 - 35847*x^10 - 76955*x^9 + 204984*x^8 + 232506*x^7 - 793350*x^6 + 139896*x^5 + 520524*x^4 + 1788924*x^3 - 3052656*x^2 + 1155456*x - 512704)
 
gp: K = bnfinit(x^18 - 9*x^17 + 192*x^15 - 216*x^14 - 2304*x^13 + 4032*x^12 + 16326*x^11 - 35847*x^10 - 76955*x^9 + 204984*x^8 + 232506*x^7 - 793350*x^6 + 139896*x^5 + 520524*x^4 + 1788924*x^3 - 3052656*x^2 + 1155456*x - 512704, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 192 x^{15} - 216 x^{14} - 2304 x^{13} + 4032 x^{12} + 16326 x^{11} - 35847 x^{10} - 76955 x^{9} + 204984 x^{8} + 232506 x^{7} - 793350 x^{6} + 139896 x^{5} + 520524 x^{4} + 1788924 x^{3} - 3052656 x^{2} + 1155456 x - 512704 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(478338621956052404233641711607222272=2^{12}\cdot 3^{44}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.99$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{1}{12} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} - \frac{1}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} + \frac{1}{8} a^{9} - \frac{5}{24} a^{8} - \frac{1}{8} a^{7} + \frac{1}{6} a^{6} + \frac{1}{4} a^{5} + \frac{1}{12} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{24} a^{15} + \frac{1}{6} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{48} a^{16} - \frac{1}{24} a^{9} + \frac{7}{48} a^{8} + \frac{1}{12} a^{7} - \frac{1}{4} a^{6} + \frac{11}{24} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{5}{12} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{11444726696638837958969876862192494164707619440} a^{17} - \frac{15989795847769827310233680840769253572344881}{3814908898879612652989958954064164721569206480} a^{16} - \frac{92444107531457645470102789462951073151162629}{5722363348319418979484938431096247082353809720} a^{15} + \frac{3064191505844267886928906072516714577701739}{248798406448670390412388627438967264450165640} a^{14} - \frac{224986729043689406036576009974007738010616711}{5722363348319418979484938431096247082353809720} a^{13} + \frac{91617315584885358566207997678455819244343387}{5722363348319418979484938431096247082353809720} a^{12} + \frac{9055218338413523965706641080100900791967897}{197322874079979964809825463141249899391510680} a^{11} + \frac{224285757243534607706732197312563028644885953}{2861181674159709489742469215548123541176904860} a^{10} + \frac{351536582803654812730017264678029421486098515}{2288945339327767591793975372438498832941523888} a^{9} - \frac{44262749613129993479065324969053446193454865}{762981779775922530597991790812832944313841296} a^{8} - \frac{17146616596480310307865535486264789942183182}{238431806179975790811872434629010295098075405} a^{7} - \frac{137859696305049924082772115288074672012479915}{1144472669663883795896987686219249416470761944} a^{6} + \frac{29519331082636961421992247353981473024063183}{104042969969443981445180698747204492406432904} a^{5} - \frac{123885497357496115457165636054388373955796593}{1430590837079854744871234607774061770588452430} a^{4} - \frac{3455122993512263300019610846607294824059337}{190745444943980632649497947703208236078460324} a^{3} + \frac{390814438545990527261698988547061723365777257}{953727224719903163247489738516041180392301620} a^{2} - \frac{78752457714731332571082872458579015367524}{238431806179975790811872434629010295098075405} a - \frac{287939375602635058392089209920430171533320371}{715295418539927372435617303887030885294226215}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 105098067902.09058 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

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 $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 3.1.243.1, 6.2.290107737.3, 9.1.2008387814976.4

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$