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Properties

Label 18.18.1015634954...2336.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{35}\cdot 3^{27}\cdot 53^{16}$
Root discriminant $681.88$
Ramified primes $2, 3, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times D_9$ (as 18T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6047664640, -18643490304, -82190351232, -77451096320, -6285328512, 21082331520, 6135260496, -2128457952, -889486704, 103129520, 58781664, -2583432, -2034564, 34980, 36729, -212, -318, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 318*x^16 - 212*x^15 + 36729*x^14 + 34980*x^13 - 2034564*x^12 - 2583432*x^11 + 58781664*x^10 + 103129520*x^9 - 889486704*x^8 - 2128457952*x^7 + 6135260496*x^6 + 21082331520*x^5 - 6285328512*x^4 - 77451096320*x^3 - 82190351232*x^2 - 18643490304*x + 6047664640)
 
gp: K = bnfinit(x^18 - 318*x^16 - 212*x^15 + 36729*x^14 + 34980*x^13 - 2034564*x^12 - 2583432*x^11 + 58781664*x^10 + 103129520*x^9 - 889486704*x^8 - 2128457952*x^7 + 6135260496*x^6 + 21082331520*x^5 - 6285328512*x^4 - 77451096320*x^3 - 82190351232*x^2 - 18643490304*x + 6047664640, 1)
 

Normalized defining polynomial

\( x^{18} - 318 x^{16} - 212 x^{15} + 36729 x^{14} + 34980 x^{13} - 2034564 x^{12} - 2583432 x^{11} + 58781664 x^{10} + 103129520 x^{9} - 889486704 x^{8} - 2128457952 x^{7} + 6135260496 x^{6} + 21082331520 x^{5} - 6285328512 x^{4} - 77451096320 x^{3} - 82190351232 x^{2} - 18643490304 x + 6047664640 \)

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:  \(1015634954431839220260170077569414080777988222222336=2^{35}\cdot 3^{27}\cdot 53^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $681.88$
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:  $2, 3, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{424} a^{9} + \frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{424} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{848} a^{11} + \frac{1}{16} a^{7} + \frac{1}{8} a^{5}$, $\frac{1}{8480} a^{12} + \frac{1}{4240} a^{11} - \frac{1}{4240} a^{10} - \frac{1}{1060} a^{9} + \frac{1}{160} a^{8} + \frac{3}{80} a^{7} - \frac{1}{20} a^{5} + \frac{9}{40} a^{4} + \frac{1}{20} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a$, $\frac{1}{16960} a^{13} - \frac{3}{8480} a^{11} + \frac{1}{1060} a^{10} - \frac{11}{16960} a^{9} + \frac{1}{80} a^{8} + \frac{7}{80} a^{7} + \frac{3}{80} a^{6} + \frac{3}{80} a^{5} + \frac{7}{40} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a$, $\frac{1}{16960} a^{14} + \frac{1}{2120} a^{11} + \frac{17}{16960} a^{10} + \frac{1}{4240} a^{9} - \frac{3}{160} a^{8} - \frac{3}{80} a^{7} + \frac{3}{80} a^{6} - \frac{9}{40} a^{5} - \frac{1}{8} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{508800} a^{15} + \frac{1}{84800} a^{14} - \frac{1}{84800} a^{13} - \frac{7}{127200} a^{12} - \frac{97}{169600} a^{11} - \frac{93}{84800} a^{10} + \frac{5}{5088} a^{9} - \frac{19}{800} a^{8} + \frac{13}{800} a^{7} - \frac{11}{300} a^{6} - \frac{7}{200} a^{5} - \frac{9}{50} a^{4} + \frac{2}{15} a^{3} - \frac{8}{25} a^{2} - \frac{12}{25} a + \frac{7}{15}$, $\frac{1}{148569600} a^{16} + \frac{1}{7428480} a^{15} + \frac{229}{12380800} a^{14} + \frac{737}{37142400} a^{13} - \frac{6623}{148569600} a^{12} + \frac{553}{1547600} a^{11} - \frac{11741}{74284800} a^{10} - \frac{10859}{9285600} a^{9} + \frac{11337}{233600} a^{8} + \frac{263}{43800} a^{7} + \frac{269}{4800} a^{6} - \frac{1623}{7300} a^{5} + \frac{13669}{87600} a^{4} - \frac{3409}{21900} a^{3} - \frac{1013}{14600} a^{2} - \frac{1037}{10950} a + \frac{259}{2190}$, $\frac{1}{39227805848114521012204035332819730624000} a^{17} - \frac{454562362848664935333068180117}{169085370034976383673293255744912632000} a^{16} - \frac{1113761805560680458424260181581781}{1225868932753578781631376104150616582000} a^{15} - \frac{135204029970407955133839559991674141}{9806951462028630253051008833204932656000} a^{14} + \frac{16462297686137465721497410579800373}{7845561169622904202440807066563946124800} a^{13} - \frac{42765933272811429091502577269937203}{1961390292405726050610201766640986531200} a^{12} + \frac{5533950391472792437679915023678968853}{19613902924057260506102017666409865312000} a^{11} - \frac{2697877305116075385253631514958824511}{2451737865507157563262752208301233164000} a^{10} + \frac{739310648405299023933305693165043407}{9806951462028630253051008833204932656000} a^{9} - \frac{1857141336788399908841652176711878939}{46259205009569010627599098269834588000} a^{8} - \frac{915546726552634694523716029128120869}{92518410019138021255198196539669176000} a^{7} + \frac{590689568703908791259226023874928687}{23129602504784505313799549134917294000} a^{6} - \frac{1082433558787463269025365269076463807}{23129602504784505313799549134917294000} a^{5} - \frac{305977092217838124486072116333902231}{11564801252392252656899774567458647000} a^{4} + \frac{1476761304851343054865650729529773233}{11564801252392252656899774567458647000} a^{3} - \frac{574079548869834022406328731815592929}{1445600156549031582112471820932330875} a^{2} - \frac{1367516960912008408974154130260964987}{2891200313098063164224943641864661750} a - \frac{3592438466900717031576271867861672}{9969656252062286773189460834016075}$

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:  \( 5212831613330000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times D_9$ (as 18T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 18 conjugacy class representatives for $S_3\times D_9$
Character table for $S_3\times D_9$

Intermediate fields

\(\Q(\sqrt{6}) \), 3.3.33708.1, 6.6.436312037376.1

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

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ R $18$

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$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
$3$3.6.9.16$x^{6} + 3 x^{4} + 6 x^{3} + 3$$6$$1$$9$$S_3^2$$[3/2, 2]_{2}^{2}$
3.12.18.78$x^{12} - 15 x^{11} - 24 x^{10} - 15 x^{9} - 9 x^{7} + 21 x^{6} + 18 x^{5} - 9 x^{4} - 36 x^{3} + 36$$6$$2$$18$$S_3^2$$[3/2, 2]_{2}^{2}$
$53$53.9.8.1$x^{9} - 53$$9$$1$$8$$D_{9}$$[\ ]_{9}^{2}$
53.9.8.1$x^{9} - 53$$9$$1$$8$$D_{9}$$[\ ]_{9}^{2}$