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Properties

Label 18.0.85804311835...2336.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 11^{9}\cdot 313^{15}$
Root discriminant $1126.84$
Ramified primes $2, 11, 313$
Class number $17945699688$ (GRH)
Class group $[9, 9, 27, 54, 151956]$ (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![25987884553, 113162419542, 219504667881, 253917378998, 199374151874, 115116872056, 51518737541, 18426951344, 5351249319, 1270005030, 245003955, 37700470, 4437353, 353158, 11784, -1382, -172, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 172*x^16 - 1382*x^15 + 11784*x^14 + 353158*x^13 + 4437353*x^12 + 37700470*x^11 + 245003955*x^10 + 1270005030*x^9 + 5351249319*x^8 + 18426951344*x^7 + 51518737541*x^6 + 115116872056*x^5 + 199374151874*x^4 + 253917378998*x^3 + 219504667881*x^2 + 113162419542*x + 25987884553)
 
gp: K = bnfinit(x^18 - 8*x^17 - 172*x^16 - 1382*x^15 + 11784*x^14 + 353158*x^13 + 4437353*x^12 + 37700470*x^11 + 245003955*x^10 + 1270005030*x^9 + 5351249319*x^8 + 18426951344*x^7 + 51518737541*x^6 + 115116872056*x^5 + 199374151874*x^4 + 253917378998*x^3 + 219504667881*x^2 + 113162419542*x + 25987884553, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 172 x^{16} - 1382 x^{15} + 11784 x^{14} + 353158 x^{13} + 4437353 x^{12} + 37700470 x^{11} + 245003955 x^{10} + 1270005030 x^{9} + 5351249319 x^{8} + 18426951344 x^{7} + 51518737541 x^{6} + 115116872056 x^{5} + 199374151874 x^{4} + 253917378998 x^{3} + 219504667881 x^{2} + 113162419542 x + 25987884553 \)

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:  \(-8580431183528775336283156521153602102325171937488142336=-\,2^{27}\cdot 11^{9}\cdot 313^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1126.84$
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:  $2, 11, 313$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{22} a^{12} + \frac{4}{11} a^{11} - \frac{9}{22} a^{10} + \frac{5}{11} a^{9} - \frac{9}{22} a^{8} + \frac{4}{11} a^{7} - \frac{5}{22} a^{6} + \frac{4}{11} a^{5} - \frac{1}{22} a^{4} - \frac{4}{11} a^{3} - \frac{5}{22} a^{2} - \frac{2}{11} a - \frac{7}{22}$, $\frac{1}{22} a^{13} - \frac{7}{22} a^{11} - \frac{3}{11} a^{10} - \frac{1}{22} a^{9} - \frac{4}{11} a^{8} - \frac{3}{22} a^{7} + \frac{2}{11} a^{6} + \frac{1}{22} a^{5} - \frac{7}{22} a^{3} - \frac{4}{11} a^{2} + \frac{3}{22} a - \frac{5}{11}$, $\frac{1}{22} a^{14} + \frac{3}{11} a^{11} + \frac{1}{11} a^{10} - \frac{2}{11} a^{9} - \frac{3}{11} a^{7} + \frac{5}{11} a^{6} - \frac{5}{11} a^{5} + \frac{4}{11} a^{4} + \frac{1}{11} a^{3} - \frac{5}{11} a^{2} + \frac{3}{11} a - \frac{5}{22}$, $\frac{1}{6886} a^{15} - \frac{7}{3443} a^{14} - \frac{50}{3443} a^{13} + \frac{2}{3443} a^{12} - \frac{1602}{3443} a^{11} + \frac{645}{3443} a^{10} + \frac{1542}{3443} a^{9} + \frac{329}{3443} a^{8} + \frac{222}{3443} a^{7} - \frac{1018}{3443} a^{6} + \frac{1468}{3443} a^{5} + \frac{540}{3443} a^{4} + \frac{1461}{3443} a^{3} + \frac{511}{3443} a^{2} - \frac{381}{6886} a + \frac{3}{3443}$, $\frac{1}{2303731958} a^{16} - \frac{2823}{209430178} a^{15} + \frac{46685517}{2303731958} a^{14} + \frac{12619632}{1151865979} a^{13} + \frac{1900395}{1151865979} a^{12} + \frac{97696129}{1151865979} a^{11} - \frac{289719132}{1151865979} a^{10} - \frac{383178083}{1151865979} a^{9} - \frac{461366679}{1151865979} a^{8} + \frac{556148687}{1151865979} a^{7} + \frac{261580302}{1151865979} a^{6} + \frac{559581663}{1151865979} a^{5} - \frac{87120747}{1151865979} a^{4} - \frac{565406045}{1151865979} a^{3} + \frac{4171835}{23749814} a^{2} - \frac{1117705999}{2303731958} a + \frac{154858629}{2303731958}$, $\frac{1}{4999725483353955587503310104056270784019181709162} a^{17} - \frac{125343911137211520920342017265020473707}{2499862741676977793751655052028135392009590854581} a^{16} + \frac{139446510256505882230072920642000673938070345}{2499862741676977793751655052028135392009590854581} a^{15} + \frac{19585654895396215698470257901082407809433526764}{2499862741676977793751655052028135392009590854581} a^{14} - \frac{25154762421017160280812473405702423011458216642}{2499862741676977793751655052028135392009590854581} a^{13} - \frac{3661801454016742511635390658299295874464635071}{227260249243361617613786822911648672000871895871} a^{12} + \frac{896826101135649056215531455471045868513807942132}{2499862741676977793751655052028135392009590854581} a^{11} - \frac{1234136180873811078425406447450883348167464890980}{2499862741676977793751655052028135392009590854581} a^{10} + \frac{1046336042950681100891074278436654990439899252038}{2499862741676977793751655052028135392009590854581} a^{9} - \frac{62092413033644139582848241404126724855204967291}{227260249243361617613786822911648672000871895871} a^{8} - \frac{395537267074489152411271504026383014428568791857}{2499862741676977793751655052028135392009590854581} a^{7} - \frac{994581344620885999827230784649445448417546140084}{2499862741676977793751655052028135392009590854581} a^{6} + \frac{684128982079721834332676485982690093393638070970}{2499862741676977793751655052028135392009590854581} a^{5} + \frac{481111188435523068681201914229974005110245661600}{2499862741676977793751655052028135392009590854581} a^{4} - \frac{73662242724986904059323879587337827635202244023}{454520498486723235227573645823297344001743791742} a^{3} - \frac{108721945506569561197997632193130396947893288093}{2499862741676977793751655052028135392009590854581} a^{2} - \frac{260414694174087498301998088656735983740601772282}{2499862741676977793751655052028135392009590854581} a - \frac{15642189495105506175739502610860573549615789760}{35209334389816588644389507775044160450839307811}$

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

Class group and class number

$C_{9}\times C_{9}\times C_{27}\times C_{54}\times C_{151956}$, which has order $17945699688$ (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:  \( -1 \) (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:  \( 5926891982.334674 \) (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

\(\Q(\sqrt{-6886}) \), 3.1.27544.1 x3, 3.3.97969.1, 6.0.20896859805184.1, Deg 6 x2, Deg 6, 9.3.200566492330693110797824.1 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
313Data not computed