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Properties

Label 18.0.31516489264...9808.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 7^{12}$
Root discriminant $43.53$
Ramified primes $2, 3, 7$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48, -3456, 86328, -243648, 311760, -214326, 48007, 23229, -13371, 10800, -8067, 3339, -524, -189, 195, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 195*x^14 - 189*x^13 - 524*x^12 + 3339*x^11 - 8067*x^10 + 10800*x^9 - 13371*x^8 + 23229*x^7 + 48007*x^6 - 214326*x^5 + 311760*x^4 - 243648*x^3 + 86328*x^2 - 3456*x + 48)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 195*x^14 - 189*x^13 - 524*x^12 + 3339*x^11 - 8067*x^10 + 10800*x^9 - 13371*x^8 + 23229*x^7 + 48007*x^6 - 214326*x^5 + 311760*x^4 - 243648*x^3 + 86328*x^2 - 3456*x + 48, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 195 x^{14} - 189 x^{13} - 524 x^{12} + 3339 x^{11} - 8067 x^{10} + 10800 x^{9} - 13371 x^{8} + 23229 x^{7} + 48007 x^{6} - 214326 x^{5} + 311760 x^{4} - 243648 x^{3} + 86328 x^{2} - 3456 x + 48 \)

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:  \(-315164892649262158461997559808=-\,2^{12}\cdot 3^{33}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.53$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{148} a^{12} - \frac{3}{74} a^{11} - \frac{5}{37} a^{10} + \frac{7}{148} a^{9} - \frac{1}{37} a^{8} + \frac{14}{37} a^{7} - \frac{29}{148} a^{6} - \frac{29}{74} a^{5} - \frac{8}{37} a^{4} - \frac{43}{148} a^{3} + \frac{3}{37} a^{2} - \frac{8}{37} a + \frac{12}{37}$, $\frac{1}{148} a^{13} + \frac{9}{74} a^{11} + \frac{35}{148} a^{10} - \frac{9}{37} a^{9} + \frac{8}{37} a^{8} + \frac{11}{148} a^{7} + \frac{16}{37} a^{6} - \frac{5}{74} a^{5} + \frac{61}{148} a^{4} - \frac{6}{37} a^{3} + \frac{10}{37} a^{2} + \frac{1}{37} a - \frac{2}{37}$, $\frac{1}{5624} a^{14} - \frac{7}{5624} a^{13} - \frac{1}{2812} a^{12} + \frac{103}{5624} a^{11} + \frac{1377}{5624} a^{10} + \frac{55}{703} a^{9} - \frac{1317}{5624} a^{8} - \frac{83}{296} a^{7} + \frac{11}{148} a^{6} - \frac{2631}{5624} a^{5} - \frac{329}{5624} a^{4} + \frac{263}{703} a^{3} + \frac{65}{703} a^{2} + \frac{225}{1406} a + \frac{35}{703}$, $\frac{1}{5624} a^{15} - \frac{13}{5624} a^{13} + \frac{13}{5624} a^{12} + \frac{213}{2812} a^{11} - \frac{1131}{5624} a^{10} - \frac{137}{5624} a^{9} - \frac{105}{703} a^{8} + \frac{127}{296} a^{7} - \frac{693}{5624} a^{6} - \frac{329}{2812} a^{5} + \frac{47}{152} a^{4} + \frac{183}{1406} a^{3} + \frac{292}{703} a^{2} - \frac{521}{1406} a - \frac{249}{703}$, $\frac{1}{2957364928376} a^{16} - \frac{1}{369670616047} a^{15} - \frac{94814931}{1478682464188} a^{14} + \frac{663704587}{1478682464188} a^{13} + \frac{599063073}{739341232094} a^{12} - \frac{15816916689}{1478682464188} a^{11} + \frac{40228831484}{369670616047} a^{10} + \frac{47785649265}{739341232094} a^{9} + \frac{6470012049}{1478682464188} a^{8} + \frac{18331661277}{39964390924} a^{7} + \frac{305671621307}{739341232094} a^{6} + \frac{684057748029}{1478682464188} a^{5} - \frac{1074182287501}{2957364928376} a^{4} + \frac{93219884925}{739341232094} a^{3} + \frac{252837576051}{739341232094} a^{2} + \frac{289880978329}{739341232094} a + \frac{48376374562}{369670616047}$, $\frac{1}{48888199630983656} a^{17} + \frac{8257}{48888199630983656} a^{16} - \frac{438185932227}{6111024953872957} a^{15} - \frac{677286403079}{24444099815491828} a^{14} + \frac{12315480618801}{6111024953872957} a^{13} + \frac{15690816450799}{6111024953872957} a^{12} - \frac{2970822740381505}{12222049907745914} a^{11} + \frac{5449844140826757}{24444099815491828} a^{10} + \frac{669713592364136}{6111024953872957} a^{9} - \frac{59313490197975}{400722947794948} a^{8} - \frac{1230326118037863}{6111024953872957} a^{7} + \frac{1838863177367565}{6111024953872957} a^{6} - \frac{5157652962630057}{48888199630983656} a^{5} - \frac{21405754701149135}{48888199630983656} a^{4} + \frac{732922642634126}{6111024953872957} a^{3} - \frac{1645862483048641}{6111024953872957} a^{2} - \frac{1400341745875785}{12222049907745914} a - \frac{117457495733209}{321632892309103}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( \frac{1783257339}{915029565602} a^{17} - \frac{30315374763}{1830059131204} a^{16} + \frac{124744279827}{1830059131204} a^{15} - \frac{658549206885}{3660118262408} a^{14} + \frac{1111934552841}{3660118262408} a^{13} - \frac{114730470036}{457514782801} a^{12} - \frac{4000629742215}{3660118262408} a^{11} + \frac{21686415985245}{3660118262408} a^{10} - \frac{23811631487759}{1830059131204} a^{9} + \frac{944062593465}{60001938728} a^{8} - \frac{75125777665425}{3660118262408} a^{7} + \frac{17132052993615}{457514782801} a^{6} + \frac{397779529103811}{3660118262408} a^{5} - \frac{1304285392559151}{3660118262408} a^{4} + \frac{208838396977517}{457514782801} a^{3} - \frac{285300118420785}{915029565602} a^{2} + \frac{75004173107379}{915029565602} a - \frac{541870814143}{457514782801} \) (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:  \( 36111792.2397 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 3.1.47628.4 x3, 3.1.588.1 x3, 3.1.47628.1 x3, 6.0.177147.2, 6.0.6805279152.3, 6.0.1037232.1, 6.0.6805279152.1, 9.1.324121835451456.8 x9

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

Sibling fields

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
$3$3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$