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Properties

Label 18.0.32216080915...6768.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 43^{12}$
Root discriminant $33.75$
Ramified primes $2, 3, 43$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3267, -14553, 36657, -67032, 93006, -78474, 39154, -14568, 10269, -7983, 5055, -1824, 382, -210, 198, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 198*x^14 - 210*x^13 + 382*x^12 - 1824*x^11 + 5055*x^10 - 7983*x^9 + 10269*x^8 - 14568*x^7 + 39154*x^6 - 78474*x^5 + 93006*x^4 - 67032*x^3 + 36657*x^2 - 14553*x + 3267)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 198*x^14 - 210*x^13 + 382*x^12 - 1824*x^11 + 5055*x^10 - 7983*x^9 + 10269*x^8 - 14568*x^7 + 39154*x^6 - 78474*x^5 + 93006*x^4 - 67032*x^3 + 36657*x^2 - 14553*x + 3267, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 198 x^{14} - 210 x^{13} + 382 x^{12} - 1824 x^{11} + 5055 x^{10} - 7983 x^{9} + 10269 x^{8} - 14568 x^{7} + 39154 x^{6} - 78474 x^{5} + 93006 x^{4} - 67032 x^{3} + 36657 x^{2} - 14553 x + 3267 \)

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:  \(-3221608091576399017168416768=-\,2^{12}\cdot 3^{9}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.75$
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:  $2, 3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} + \frac{5}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{7} - \frac{1}{6} a^{5} - \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{72} a^{12} - \frac{1}{72} a^{10} + \frac{5}{72} a^{6} + \frac{1}{4} a^{5} - \frac{5}{18} a^{4} + \frac{1}{4} a^{3} - \frac{7}{24} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{72} a^{13} - \frac{1}{72} a^{11} + \frac{5}{72} a^{7} - \frac{1}{12} a^{6} - \frac{5}{18} a^{5} - \frac{1}{12} a^{4} - \frac{7}{24} a^{3} - \frac{1}{12} a^{2} - \frac{3}{8} a$, $\frac{1}{3168} a^{14} - \frac{7}{3168} a^{13} + \frac{1}{176} a^{12} - \frac{17}{3168} a^{11} - \frac{37}{3168} a^{10} - \frac{5}{176} a^{9} + \frac{29}{3168} a^{8} - \frac{5}{3168} a^{7} - \frac{7}{1056} a^{6} + \frac{745}{1584} a^{5} - \frac{263}{3168} a^{4} + \frac{41}{1056} a^{3} - \frac{13}{528} a^{2} + \frac{137}{352} a - \frac{7}{32}$, $\frac{1}{136224} a^{15} + \frac{7}{68112} a^{14} - \frac{833}{136224} a^{13} - \frac{739}{136224} a^{12} - \frac{1561}{68112} a^{11} - \frac{1879}{136224} a^{10} - \frac{5557}{136224} a^{9} - \frac{377}{34056} a^{8} - \frac{2219}{68112} a^{7} + \frac{1357}{136224} a^{6} + \frac{59627}{136224} a^{5} - \frac{1093}{17028} a^{4} - \frac{16765}{45408} a^{3} - \frac{18307}{45408} a^{2} - \frac{475}{1892} a - \frac{135}{1376}$, $\frac{1}{77766467328} a^{16} - \frac{1}{9720808416} a^{15} - \frac{2067349}{19441616832} a^{14} + \frac{18697}{25118368} a^{13} + \frac{141467581}{38883233664} a^{12} - \frac{38283253}{1215101052} a^{11} - \frac{1441657}{147284976} a^{10} + \frac{9789445}{441854928} a^{9} + \frac{2761990591}{77766467328} a^{8} + \frac{1983039}{135011228} a^{7} + \frac{140840731}{4860404208} a^{6} - \frac{2593370851}{9720808416} a^{5} + \frac{1796488981}{38883233664} a^{4} + \frac{1519637279}{3240269472} a^{3} - \frac{917323003}{3240269472} a^{2} - \frac{31991631}{1080089824} a + \frac{245854395}{785519872}$, $\frac{1}{4588221572352} a^{17} + \frac{7}{1529407190784} a^{16} + \frac{357463}{382351797696} a^{15} - \frac{69030113}{1147055393088} a^{14} + \frac{1778570119}{764703595392} a^{13} + \frac{649783483}{208555526016} a^{12} + \frac{8837553497}{573527696544} a^{11} + \frac{1639264039}{52138881504} a^{10} + \frac{20245833879}{509802396928} a^{9} + \frac{60950595851}{4588221572352} a^{8} - \frac{1764144355}{63725299616} a^{7} + \frac{1010512499}{13034720376} a^{6} - \frac{240257548615}{2294110786176} a^{5} + \frac{7508570075}{38883233664} a^{4} + \frac{48121670207}{191175898848} a^{3} + \frac{487838964}{1991415613} a^{2} - \frac{67833839239}{509802396928} a - \frac{1021703833}{46345672448}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{1049318}{1991415613} a^{17} + \frac{8919203}{1991415613} a^{16} - \frac{326409544}{17922740517} a^{15} + \frac{280871680}{5974246839} a^{14} - \frac{1393520716}{17922740517} a^{13} + \frac{382702684}{5974246839} a^{12} - \frac{2811832240}{17922740517} a^{11} + \frac{474114896}{543113349} a^{10} - \frac{39374016410}{17922740517} a^{9} + \frac{17657829122}{5974246839} a^{8} - \frac{64141032752}{17922740517} a^{7} + \frac{32397695960}{5974246839} a^{6} - \frac{28233308788}{1629340047} a^{5} + \frac{293186240}{9205311} a^{4} - \frac{179252247200}{5974246839} a^{3} + \frac{30553397024}{1991415613} a^{2} - \frac{13855403030}{1991415613} a + \frac{430824325}{181037783} \) (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:  \( 2928309.61291 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.22188.1 x3, 3.3.1849.1, 6.0.1476922032.1, 6.0.92307627.1, 6.0.798768.2 x2, 9.3.10923315348672.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.798768.2
Degree 9 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$43$43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$