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Properties

Label 18.0.51956851764...4963.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 1129^{8}$
Root discriminant $39.38$
Ramified primes $3, 1129$
Class number $76$ (GRH)
Class group $[2, 38]$ (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, 60, 3678, -4564, 9457, -8321, 11934, -9256, 9392, -5541, 4263, -2046, 1289, -457, 209, -46, 19, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 19*x^16 - 46*x^15 + 209*x^14 - 457*x^13 + 1289*x^12 - 2046*x^11 + 4263*x^10 - 5541*x^9 + 9392*x^8 - 9256*x^7 + 11934*x^6 - 8321*x^5 + 9457*x^4 - 4564*x^3 + 3678*x^2 + 60*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 19*x^16 - 46*x^15 + 209*x^14 - 457*x^13 + 1289*x^12 - 2046*x^11 + 4263*x^10 - 5541*x^9 + 9392*x^8 - 9256*x^7 + 11934*x^6 - 8321*x^5 + 9457*x^4 - 4564*x^3 + 3678*x^2 + 60*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 19 x^{16} - 46 x^{15} + 209 x^{14} - 457 x^{13} + 1289 x^{12} - 2046 x^{11} + 4263 x^{10} - 5541 x^{9} + 9392 x^{8} - 9256 x^{7} + 11934 x^{6} - 8321 x^{5} + 9457 x^{4} - 4564 x^{3} + 3678 x^{2} + 60 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-51956851764606118870683564963=-\,3^{9}\cdot 1129^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.38$
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:  $3, 1129$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} - \frac{4}{19} a^{14} + \frac{1}{19} a^{13} + \frac{2}{19} a^{12} - \frac{1}{19} a^{11} - \frac{7}{19} a^{10} + \frac{5}{19} a^{9} + \frac{4}{19} a^{8} - \frac{5}{19} a^{7} - \frac{5}{19} a^{6} + \frac{3}{19} a^{5} - \frac{5}{19} a^{4} + \frac{3}{19} a^{3} - \frac{9}{19} a^{2} + \frac{1}{19}$, $\frac{1}{19} a^{16} + \frac{4}{19} a^{14} + \frac{6}{19} a^{13} + \frac{7}{19} a^{12} + \frac{8}{19} a^{11} - \frac{4}{19} a^{10} + \frac{5}{19} a^{9} - \frac{8}{19} a^{8} - \frac{6}{19} a^{7} + \frac{2}{19} a^{6} + \frac{7}{19} a^{5} + \frac{2}{19} a^{4} + \frac{3}{19} a^{3} + \frac{2}{19} a^{2} + \frac{1}{19} a + \frac{4}{19}$, $\frac{1}{1780344773928222178369891} a^{17} + \frac{4969940851627676409236}{1780344773928222178369891} a^{16} + \frac{729900469801086641213}{61391199100973178564479} a^{15} - \frac{534260070876636498669762}{1780344773928222178369891} a^{14} - \frac{697248843309388310159185}{1780344773928222178369891} a^{13} + \frac{154251296482725023996580}{1780344773928222178369891} a^{12} - \frac{130433909560618703932708}{1780344773928222178369891} a^{11} - \frac{399079801231151187378432}{1780344773928222178369891} a^{10} + \frac{14394465569102650703129}{1780344773928222178369891} a^{9} - \frac{367998019632250108301181}{1780344773928222178369891} a^{8} - \frac{34115077627552824437861}{93702356522538009387889} a^{7} + \frac{511361464584614771527982}{1780344773928222178369891} a^{6} - \frac{416332898405329642739923}{1780344773928222178369891} a^{5} + \frac{303603024426334874727568}{1780344773928222178369891} a^{4} - \frac{26773299653594153180341}{93702356522538009387889} a^{3} + \frac{478270407392797804205701}{1780344773928222178369891} a^{2} - \frac{870696502220519743050815}{1780344773928222178369891} a + \frac{131112507893766808919788}{1780344773928222178369891}$

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

Class group and class number

$C_{2}\times C_{38}$, which has order $76$ (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:  \( -\frac{29333510837476704074339}{1780344773928222178369891} a^{17} + \frac{88213177518867648875147}{1780344773928222178369891} a^{16} - \frac{19222870878466126699084}{61391199100973178564479} a^{15} + \frac{1351885121027244340292883}{1780344773928222178369891} a^{14} - \frac{6131174220851336992783571}{1780344773928222178369891} a^{13} + \frac{13428241015122774316752373}{1780344773928222178369891} a^{12} - \frac{37809088739123548025245491}{1780344773928222178369891} a^{11} + \frac{60082497022956651666520939}{1780344773928222178369891} a^{10} - \frac{124909412125943900374206092}{1780344773928222178369891} a^{9} + \frac{162620973139084496762883691}{1780344773928222178369891} a^{8} - \frac{14474050186287256946169357}{93702356522538009387889} a^{7} + \frac{271601669308298606333765854}{1780344773928222178369891} a^{6} - \frac{348723085871212613963280841}{1780344773928222178369891} a^{5} + \frac{244123161069096676601877974}{1780344773928222178369891} a^{4} - \frac{14530595680273679248173192}{93702356522538009387889} a^{3} + \frac{135441400744220539858470835}{1780344773928222178369891} a^{2} - \frac{106849345692613212166805757}{1780344773928222178369891} a + \frac{37294591531552648308581}{1780344773928222178369891} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 290542.983381 \) (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

\(\Q(\sqrt{-3}) \), 3.3.1129.1, 6.0.34415307.1, 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 $18$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1129Data not computed