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Properties

Label 18.0.48163618447...7952.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 37^{14}\cdot 103^{9}$
Root discriminant $267.19$
Ramified primes $2, 37, 103$
Class number $436101120$ (GRH)
Class group $[2, 2, 2, 2, 2, 208, 65520]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13377444091051, -3662150838941, 3940350373970, -856272430360, 506133264109, -89558691519, 38018012120, -5538379279, 1870301948, -224224367, 63312072, -6163867, 1489716, -113727, 23619, -1298, 228, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 228*x^16 - 1298*x^15 + 23619*x^14 - 113727*x^13 + 1489716*x^12 - 6163867*x^11 + 63312072*x^10 - 224224367*x^9 + 1870301948*x^8 - 5538379279*x^7 + 38018012120*x^6 - 89558691519*x^5 + 506133264109*x^4 - 856272430360*x^3 + 3940350373970*x^2 - 3662150838941*x + 13377444091051)
 
gp: K = bnfinit(x^18 - 7*x^17 + 228*x^16 - 1298*x^15 + 23619*x^14 - 113727*x^13 + 1489716*x^12 - 6163867*x^11 + 63312072*x^10 - 224224367*x^9 + 1870301948*x^8 - 5538379279*x^7 + 38018012120*x^6 - 89558691519*x^5 + 506133264109*x^4 - 856272430360*x^3 + 3940350373970*x^2 - 3662150838941*x + 13377444091051, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 228 x^{16} - 1298 x^{15} + 23619 x^{14} - 113727 x^{13} + 1489716 x^{12} - 6163867 x^{11} + 63312072 x^{10} - 224224367 x^{9} + 1870301948 x^{8} - 5538379279 x^{7} + 38018012120 x^{6} - 89558691519 x^{5} + 506133264109 x^{4} - 856272430360 x^{3} + 3940350373970 x^{2} - 3662150838941 x + 13377444091051 \)

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:  \(-48163618447522673224353606318670413146877952=-\,2^{12}\cdot 37^{14}\cdot 103^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $267.19$
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:  $2, 37, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{17} - \frac{4174292033036070138615089897784131172161101924375754009190031332921981092}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{16} - \frac{7805274542154520213958515917376690341798655858903923030878436455583669077}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{15} + \frac{9652588484009775737728066705387867407227091883413470918301773661008295064}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{14} + \frac{4779466757882480853339425979445545688004243039716154172323034122300198819}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{13} - \frac{3226381881031734554272998027972344994654795151569276190590563339066749783}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{12} - \frac{3293828469459981809841258166568575723784857524567776091923647077063022381}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{11} - \frac{9579265923659635051319290165920343917022190775369911716365839364062361820}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{10} - \frac{2288862576457678104656507627782521383467317217048538623749072391976064768}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{9} + \frac{9976456727258875481765302137369916170821071677543881905329760005486811398}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{8} + \frac{2717224437649494892150490863251800898935020639933914737896609017678950323}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{7} - \frac{7351307269727809565256921383264708277294969876620900114513605878188735509}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{6} + \frac{8905969036984199363576902420333591392282728939051624686361295258874681935}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{5} + \frac{7579864001673806206863815016307353216690431694836072625134884483013881048}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{4} + \frac{11027151248311293082807225610603521676144109007944308344223357508289649907}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{3} - \frac{9974217279444510451814037134797997491460089556649537538820732131008501791}{25725847456625775483413913687378737454585963527701147648635750149031884299} a^{2} + \frac{12561230712509318012437429200259524010629718590012432083984199807467732802}{25725847456625775483413913687378737454585963527701147648635750149031884299} a + \frac{3791202957891810737832908349069805545578245491053104930846908857254024}{15811830028657514126253173747620613063666849125815087675867086754168337}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{208}\times C_{65520}$, which has order $436101120$ (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:  \( 615797.1340659427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-103}) \), 3.3.1369.1, 3.3.148.1, 6.0.2047946327047.1, 6.0.23935092208.2, 9.9.6075640136512.1

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 12 sibling: 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$103$103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$